Space-Time Point Pattern Simulation, Visualisation and Analysis

Authors: Edith Gabriel [aut, cre],Peter J Diggle [aut],Barry Rowlingson [aut],Francisco J Rodriguez-Cortes [aut]

Version: 2.0-1

License: GPL-3

Description

Many of the models encountered in applications of point process methods to the study of spatio-temporal phenomena are covered in ‘stpp’. This package provides statistical tools for analyzing the global and local second-order properties of spatio-temporal point processes, including estimators of the space-time inhomogeneous K-function and pair correlation function. It also includes tools to get static and dynamic display of spatio-temporal point patterns. See Gabriel et al. (2013) doi:10.18637/jss.v053.i02.

Depends

R (>= 3.4.2), splancs, KernSmooth, spatstat, rpanel, rgl, plot3D, ggplot2, gridExtra

Suggests

packagedocs

Datasets available by data()

fmd

2001 food-and-mouth epidemic, north Cumbria (UK)

This data set gives the spatial locations and reported times of food-and-mouth disease in north Cumbria (UK), 2001. It is of no scientific value, as it deliberately excludes confidential information on farms at risk in the study-region. It is included in the package purely as an illustrative example.

Usage

data(fmd)

Format

A matrix containing (x,y,t) coordinates of the 648 observations.

References

Diggle, P., Rowlingson, B. and Su, T. (2005). Point process methodology for on-line spatio-temporal disease surveillance. Environmetrics, 16, 423–34.

northcumbria

Polygon boundary of north Cumbria

This data set gives the boundary of the county of north Cumbria (UK).

Usage

data(northcumbria)

Format

A matrix containing \((x,y)\) coordinates of the boundary.

High-Level Plots

stan

(3D) space-time data animation

Displays \((x,y,t)\) point data and enables dynamic highlighting of time slices.

Usage

stan(xyt,tlim=range(xyt[,3],na.rm=TRUE),twid=diff(tlim)/20,
persist=FALSE,states,bgpoly,bgframe=TRUE,bgimage,
bgcol=gray(seq(0,1,len=12)),axes=TRUE)

Arguments

xyt: A 3-column matrix of \(x\), \(y\), \(t\) coordinates
tlim: A two-element vector of upper and lower time limits
twid: The initial time window width
persist: Whether to display points before time window
states: How to display points - see Details
bgpoly: A polygon to draw on the background plane
bgframe: Whether to extend the bgpoly to the front plane
bgimage: An list with \(x\), \(y\) vectors and \(z\) matrix to display on the background plane
bgcol: A colour palette vector with which to draw the bgimage
axes: Logical value indicating whether labels should be added.

Details

This function requires the rpanel and rgl packages. It uses rpanel for the sliders to control the graphics, and rgl for its ability to do flicker-free graphics.

The sliders set the position and width of the temporal highlight window. For time slider set to time \(T\) and width slider set to \(S\), highlighted points are those with time coordinate \(t\) such that \(T-S< t< T\).

How points are shown is configured with the states parameter. This is a list of length 3 specifying how points before the time window, inside the time window, and after the time window are displayed. Each element is a list of parameters as would be passed to material3d() together with a radius element. Points are drawn as spheres with the corresponding material and radius as a fraction of the spatial span of the data.

By default the third state is invisible, and the first two states are different. By calling with the default for states and persist=TRUE, then the first state is set to the same as the second state. This has the effect of showing all points at time \(< T\) with the same sphere type.

If the user specifies the states parameter, then persist is ignored. The user can emulate the persist behaviour by specifying a states list with identical parameters for states 1 and 2.

Note that each state element should specify all material3d parameters used in any of the state elements. This is to make sure the parameters are reset for each of the sets of points.

The background polygon must be a simple 2-column vector of \(x\) and \(y\) coordinates. When used with bgframe=TRUE, the polygon is also drawn on the front plane, and the convex hull points are connected front to back in order to visualise the space-time prism that the data are contained in.

A raster image can be displayed on the back plane by setting the bgimage parameter. This must be a list with \(x\), \(y\) and \(z\) components as needed by the image function. Note that \(x\) and \(y\) define the center of cells and so must be the same length as the dimensions of \(z\) - the image function can accept \(x\) and \(y\) values that are one longer than the dimensions of \(z\) to define the edges, but bgimage does not allow that.

Value

A list of the slider parameters when the dialog is quitted.

Author

Barry Rowlingson b.rowlingson@lancaster.ac.uk, Edith Gabriel

Package Functions

animation

Space-time data animation

Provide an animation of spatio-temporal point patterns.

Usage

animation(xyt, s.region, t.region, runtime=1, incident="red", 
prevalent="pink3", pch=19, cex=0.5, plot.s.region=TRUE, 
scales=TRUE, border.frac=0.05, add=FALSE)

Arguments

xyt: Data-matrix containing the \((x,y,t)\)-coordinates.
s.region: Two-column matrix specifying polygonal region containing all data-locations xyt[,1:2]. If missing, s.region is the bounding box of xyt[,1:2].
t.region: Interval containing all data-times xyt[,3]. If missing, t.region is defined by the range of xyt[,3].
runtime: Approximate running time of animation, in seconds, but it is longer than expected. Can also be NULL.
incident: Colour in which incident point xyt[i,1:2] is plotted at time xyt[i,3].
prevalent: Colour to which prevalent point xyt[i,1:2] fades at time xyt[i+1,3].
pch: Plotting symbol used for each point.
cex: Magnification of plotting symbol relative to standard size.
plot.s.region: If TRUE, plot s.region as polygon.
scales: If TRUE, plot X and Y axes with scales.
border.frac: Extent of border of plotting region surounding s.region, as fraction of ranges of X and Y.
add: If TRUE, add the animation to an existing plot.

Value

None

Author

Peter J Diggle, Edith Gabriel edith.gabriel@univ-avignon.fr.

as.3dpoint

Create data in spatio-temporal point format

Create data in spatio-temporal point format.

Usage

as.3dpoints(...)

Arguments

…: Any object(s), such as \(x\), \(y\) and \(t\) vectors of the same length, or a list or data frame containing \(x\), \(y\) and \(t\) vectors. Valid options for … are: a points object: returns it unaltered; a list with \(x\), \(y\) and \(t\) elements of the same length: returns a points object with the \(x\), \(y\) and \(t\) elements as the coordinates of the points; three vectors of equal length: returns a points object with the first vector as the \(x\) coordinates, the second vector as the \(y\)-coordinates and the third vector as the \(t\)-coordinates.

Value

The output is an object of the class stpp. as.3dpoints tries to return the argument(s) as a spatio-temporal points object.

Author

Edith Gabriel edith.gabriel@univ-avignon.fr, Peter Diggle, Barry Rowlingson.

ASTIKhat

Anisotropic Space-Time Inhomogeneous \(K\)-function

Compute an estimation of the Anisotropic Space-Time inhomogeneous \(K\)-function.

Usage

ASTIKhat(xyt, s.region, t.region, lambda, dist, times, ang, correction = "border")

Arguments

xyt: Coordinates and times \((x,y,t)\) of the point pattern.
s.region: Two-column matrix specifying polygonal region containing all data locations. If s.region is missing, the bounding box of xyt[,1:2] is considered.
t.region: Vector containing the minimum and maximum values of the time interval. If t.region is missing, the range of xyt[,3] is considered.
dist: Vector of distances \(u\) at which \(K_\phi(r,t)\) is computed. If missing, the maximum of dist is given by \(min(S_x, S_y)/4\), where \(S_x\) and \(S_y\) represent the maximum width and height of the bounding box of s.region.
times: Vector of times \(v\) at which \(K_\phi(r,t)\) is computed. If missing, the maximum of times is given by \((T_max - T_min)/4\), where \(T_min\) and \(T_max\) are the minimum and maximum of the time interval \(T\).
lambda: Vector of values of the space-time intensity function evaluated at the points \((x,y,t)\) in \(S x T\). If lambda is missing, the estimate of the anisotropic space-time \(K\)-function is computed as for the homogeneous case, i.e. considering \(n/|S x T|\) as an estimate of the space-time intensity.
ang: Angle in radians at which \(K_\phi(r,t)\) is computed. The argument ang=2*pi by default.
correction: A character vector specifying the edge correction(s) to be applied among “border”, “modified.border”, “translate” and “none” (see STIKhat). The default is “border”.

Value

AKhat: ndist x ntimes matrix containing values of \(K_\phi(u,v)\).
dist, times: Parameters passed in argument.
correction: The name(s) of the edge correction method(s) passed in argument.

References

Illian, J. B., Penttinen, A., Stoyan, H. and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. John Wiley and Sons, London.

Gonzalez, J. A., Rodriguez-Cortes, F. J., Cronie, O., Mateu, J. (2016). Spatio-temporal point process statistics: a review. Spatial Statistics. Accepted.

Ohser, J. and D. Stoyan (1981). On the second-order and orientation analysis of planar stationary point processes. Biometrical Journal 23, 523-533.

Author

Francisco J. Rodriguez-Cortes cortesf@uji.es

is.3dpoints

Spatio-temporal point objects

Tests for data in spatio-temporal point format.

Usage

is.3dpoints(x)

Arguments

x: any object.

Value

is.3dpoints returns TRUE if x is a spatio-temporal points object, FALSE otherwise.

Author

Edith Gabriel edith.gabriel@univ-avignon.fr, Peter Diggle, Barry Rowlingson.

LISTAhat

Estimation of the Space-Time Inhomogeneous Pair Correlation LISTA functions

Compute an estimate of the space-time pair correlarion LISTA functions.

Usage

LISTAhat(xyt, s.region, t.region, dist, times, lambda,
ks = "box", hs, kt = "box", ht, correction = "isotropic")

Arguments

xyt: Coordinates and times \((x,y,t)\) of the point pattern.
s.region: Two-column matrix specifying polygonal region containing all data locations. If s.region is missing, the bounding box of xyt[,1:2] is considered.
t.region: Vector containing the minimum and maximum values of the time interval. If t.region is missing, the range of xyt[,3] is considered.
dist: Vector of distances \(u\) at which \(g^{(i)}(u,v)\) is computed. If missing, the maximum of dist is given by \(min(S_x, S_y)/4\), where \(S_x\) and \(S_y\) represent the maximum width and height of the bounding box of s.region.
times: Vector of times \(v\) at which \(g^{(i)}(u,v)\) is computed. If missing, the maximum of times is given by \((T_max - T_min)/4\), where \(T_min\) and \(T_max\) are the minimum and maximum of the time interval \(T\).
lambda: Vector of values of the space-time intensity function evaluated at the points \((x,y,t)\) in \(S x T\). If lambda is missing, the estimate of the space-time pair correlation function is computed as for the homogeneous case, i.e. considering \((n-1)/|S x T|\) as an estimate of the space-time intensity.
ks: Kernel function for the spatial distances. Default is the “box” kernel. Can also be “epanech” for the Epanechnikov kernel or “gaussian” or “biweight”.
hs: Bandwidth of the kernel function ks.
kt: Kernel function for the temporal distances. Default is the “box” kernel. Can also be “epanech” for the Epanechnikov kernel or “gaussian” or “biweight”.
ht: Bandwidth of the kernel function kt.
correction: A character vector specifying the edge correction(s) to be applied among “isotropic”, “border”, “modified.border”, “translate” and “none” (see PCFhat). The default is “isotropic”.

Details

An individual product density LISTA functions \(g^{(i)}(.,.)\) should reveal the extent of the contribution of the event \((u_i,t_i)\) to the global estimator of the pair correlation function \(g(.,.)\), and may provide a further description of structure in the data (e.g., determining events with similar local structure through dissimilarity measures of the individual LISTA functions), for more details see Siino et al. (2017).

Value

list.LISTA: A list containing the values of the estimation of \(g^{(i)}(r,t)\) for each one of \(n\) points of the point pattern by matrixs.
listatheo: ndist x ntimes matrix containing theoretical values for a Poisson process.
dist, times: Parameters passed in argument.
kernel: Vector of names and bandwidths of the spatial and temporal kernels.
correction: The name(s) of the edge correction method(s) passed in argument.

References

Baddeley, A. and Turner, J. (2005). spatstat: An R Package for Analyzing Spatial Point Pattens. Journal of Statistical Software 12, 1-42.

Cressie, N. and Collins, L. B. (2001). Analysis of spatial point patterns using bundles of product density LISA functions. Journal of Agricultural, Biological, and Environmental Statistics 6, 118-135.

Cressie, N. and Collins, L. B. (2001). Patterns in spatial point locations: Local indicators of spatial association in a minefield with clutter Naval Research Logistics (NRL), John Wiley & Sons, Inc. 48, 333-347.

Siino, M., Rodriguez-Cortes, F. J., Mateu, J. and Adelfio, G. (2017). Testing for local structure in spatio-temporal point pattern data. Environmetrics. DOI: 10.1002/env.2463.

Stoyan, D. and Stoyan, H. (1994). Fractals, random shapes, and point fields: methods of geometrical statistics. Chichester: Wiley.

Author

Francisco J. Rodriguez-Cortes cortesf@uji.es

PCFhat

Estimation of the Space-Time Inhomogeneous Pair Correlation function

Compute an estimate of the space-time pair correlation function.

Usage

PCFhat(xyt, s.region, t.region, dist, times, lambda,
ks="box", hs, kt="box", ht, correction = "isotropic")

Arguments

xyt: Coordinates and times \((x,y,t)\) of the point pattern.
s.region: Two-column matrix specifying polygonal region containing all data locations. If s.region is missing, the bounding box of xyt[,1:2] is considered.
t.region: Vector containing the minimum and maximum values of the time interval. If t.region is missing, the range of xyt[,3] is considered.
dist: Vector of distances \(u\) at which \(g(u,v)\) is computed. If missing, the maximum of dist is given by \(min(S_x, S_y)/4\), where \(S_x\) and \(S_y\) represent the maximum width and height of the bounding box of s.region.
times: Vector of times \(v\) at which \(g(u,v)\) is computed. If missing, the maximum of times is given by \((T_max - T_min)/4\), where \(T_min\) and \(T_max\) are the minimum and maximum of the time interval \(T\).
lambda: Vector of values of the space-time intensity function evaluated at the points \((x,y,t)\) in \(S x T\). If lambda is missing, the estimate of the space-time pair correlation function is computed as for the homogeneous case, i.e. considering \(n/|S x T|\) as an estimate of the space-time intensity.
ks: Kernel function for the spatial distances. Default is the “box” kernel. Can also be “epanech” for the Epanechnikov kernel or “gaussian” or “biweight”.
hs: Bandwidth of the kernel function ks.
kt: Kernel function for the temporal distances. Default is the “box” kernel. Can also be “epanech” for the Epanechnikov kernel or “gaussian” or “biweight”.
ht: Bandwidth of the kernel function kt.
correction: A character vector specifying the edge correction(s) to be applied among “isotropic”, “border”, “modified.border”, “translate” and “none” (see Details). The default is “isotropic”.

Details

An approximately unbiased estimator for the space-time pair correlation function, based on data giving the locations of events \(x_i: i = 1,...,n\) on a spatio-temporal region \(SxT\), where \(S\) is an arbitrary polygon and \(T\) a time interval: \[\widehat{g}(u,v)=\frac{1}{4\pi u}\sum_{i=1}^{n}\sum_{j \neq i} \frac{1}{w_{ij}}\frac{k_{s}(u-\|s_i-s_j\|)k_{t}(v-|t_i-t_j|)}{\lambda(x_i) \lambda(x_j)},\] where \(lambda(x_i)\) is the intensity at \(x_i = (s_i, t_i)\) and \(w_ij\) is an edge correction factor to deal with spatial-temporal edge effects. The edge correction methods implemented are:

isotropic: \(w_ij = |S x T| w_ij^(s) w_ij^(t)\), where the temporal edge correction factor \(w_ij^(t) = 1\) if both ends of the interval of length \(2|t_i - t_j|\) centred at \(t_i\) lie within \(T\) and \(w_ij^(t) = 1/2\) otherwise and \(w_ij^(s)\) is the proportion of the circumference of a circle centred at the location \(s_i\) with radius \(||s_i - s_j||\) lying in \(S\) (also called Ripleys edge correction factor).

border: \(w_ij = (sum_{j = 1,...,n} 1{d(s_j, S) > u ; d(t_j, T) > v}/ lambda(x_j)) / 1{d(s_i, S) > u ; d(t_i, T) > v}\), where \(d(s_i, S)\) denotes the distance between \(s_i\) and the boundary of \(S\) and \(d(t_i, T)\) the distance between \(t_i\) and the boundary of \(T\).

modified.border: \(w_ij = |S_(-u) x T_(-v)| / 1{d(s_i, S) > u ; d(t_i, T) > v}\), where \(S_(-u)\) and \(T_(-v)\) are the eroded spatial and temporal region respectively, obtained by trimming off a margin of width \(u\) and \(v\) from the border of the original region.

translate: \(w_ij = |S intersect S_(s_i - s_j) x T intersect T_(t_i - t_j)|\), where \(S_(s_i - s_j)\) and \(T_(t_ i - t_j)\) are the translated spatial and temporal regions.

none: No edge correction is performed and \(w_ij = |S x T|\).

\(k_s()\) and \(k_t()\) denotes kernel functions with bandwidth \(h_s\) and \(h_t\). Experience with pair correlation function estimation recommends box kernels (the default), see Illian et al. (2008). Epanechnikov, Gaussian and biweight kernels are also implemented. Whatever the kernel function, if the bandwidth is missing, a value is obtain from the function dpik of the package KernSmooth. Note that the bandwidths play an important role and their choice is crucial in the quality of the estimators as they heavily influence their variance.

Value

pcf: ndist x ntimes matrix containing values of \(g(u,v)\).
pcftheo: ndist x ntimes matrix containing theoretical values for a Poisson process.
dist, times: Parameters passed in argument.
kernel: A vector of names and bandwidths of the spatial and temporal kernels.
correction: The name(s) of the edge correction method(s) passed in argument.

References

Baddeley, A., Rubak, E., Turner, R., (2015). Spatial Point Patterns: Methodology and Applications with R. CRC Press, Boca Raton.

Gabriel E., Diggle P. (2009). Second-order analysis of inhomogeneous spatio-temporal point process data. Statistica Neerlandica, 63, 43–51.

Gabriel E., Rowlingson B., Diggle P. (2013). stpp: an R package for plotting, simulating and analyzing Spatio-Temporal Point Patterns. Journal of Statistical Software, 53(2), 1–29.

Gabriel E. (2014). Estimating second-order characteristics of inhomogeneous spatio-temporal point processes: influence of edge correction methods and intensity estimates. Methodology and computing in Applied Probabillity, 16(2), 411–431.

Illian JB, Penttinen A, Stoyan H and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. John Wiley and Sons, London.

Author

Edith Gabriel edith.gabriel@univ-avignon.fr

plot.stpp: plot.stpp, plot

Plot for spatio-temporal point objects

This function plot either \(xy\)-locations and cumulative distribution of the times, or a space-time 3D scatter, or the time-mark and space-mark of the spatio-temporal point pattern, through arguments style and type. It can also plot \(xy\)-locations with time treated as a quantitative mark attached to each location, as in the previous version of the function, through argument mark (see stpp version < 2.0.0).

Usage

"plot"(x, s.region=NULL, t.region=NULL, style="generic", type="projection",
mark=NULL , mark.cexmin=0.4, mark.cexmax=1.2, mark.col=1, ...)

Arguments

x: Any object of class stpp in spatio-temporal point format.
s.region: Two-column matrix specifying polygonal region containing all data locations. If s.region is missing, the default limits are considered.
t.region: Vector containing the minimum and maximum values of the time interval. If t.region is missing, the default limits are considered.
type: Specify the kind of graphical representation. If type=“projection” (default) the function plot the xy-locations and cumulative distribution of the times. If type=“mark” the function plot the time-mark and space-mark. If type=“scatter” the function plot space-time 3D scatter.
style: Two different classes of graphic styles can be chosen. If style=“generic” (default) the graphics are plot by default function plot in R and if type=“elegant” the graphics are plot based on the R packages ggplot2 and plot3D.
mark: Logical. If NULL (default), xy-locations and cumulative distribution of the times are plotted. If TRUE, the time is treated as a quantitative mark attached to each location, and the locations are plotted with the size and/or colour of the plotting symbol determined by the value of the mark.
mark.cexmin, mark.cexmax: Range of the size of the plotting symbol when mark=TRUE.
mark.col: Colour of the plotting symbol when mark=TRUE. If mark.col=0, all locations have the same colour specified by the usual col argument. Otherwise, can be 1 or “black” (default), 2 or “red”, 3 or “green”, 4 or “blue”, in which cases symbols colour is faded, and the darker corresponds to the most recent time.
…: Further arguments to be passed to the functions plot and scatter3D. Typical arguments are pch, theta and phi.

Value

References

Gabriel E., Rowlingson B., Diggle P. (2013). stpp: an R package for plotting, simulating and analyzing Spatio-Temporal Point Patterns. Journal of Statistical Software, 53(2), 1–29.

Stoyan, D., Rodriguez-Cortes, F. J., Mateu, J., and Gille, W. (2017). Mark variograms for spatio-temporal point processes. Spatial Statistics. 20, 125-147.

Author

Edith Gabriel edith.gabriel@univ-avignon.fr and Francisco J. Rodriguez-Cortes.

plotK

Plot the estimation of the Space-Time Inhomogeneous K-function

Contour plot or perspective plot or image of the Space-Time Inhomogeneous K-function estimate.

Usage

plotK(K,n=15,L=FALSE,type="contour",legend=TRUE,which=NULL,
main=NULL,...)

Arguments

K: Result of the STIKhat function.
n: Number of contour levels desired.
L: Logical indicating whether \(K(u,v)\) or \(K(u,v)-pi u^2 v\) must be plotted.
type: Specifies the kind of plot: contour by default, but can also be persp or image
legend: Logical indicating whether a legend must be added to the plot.
which: A character specifying the edge correction among the ones used in STIKhat. If a single edge correction method was used in STIKhat, it is not necessary to specify which.
main: Plot title.
…: Additional arguments to persp if persp=TRUE, such as theta and phi.

Author

Edith Gabriel edith.gabriel@univ-avignon.fr

plotPCF

Plot the estimation of the Space-Time Inhomogeneous Pair Correlation function

Contour, image or perspective plot of the Space-Time Inhomogeneous Pair correlation function estimate.

Usage

plotPCF(PCF,n=15,type="contour",legend=TRUE,which=NULL,
main=NULL,...)

Arguments

PCF: Result of the PCFhat function.
n: Number of contour levels desired.
type: Specifies the kind of plot: contour by default, but can also be persp or image
legend: Logical indicating whether a legend must be added to the plot.
which: A character specifying the edge correction among the ones used in PCFhat. If a single edge correction method was used in PCFhat, it is not necessary to specify which.
main: Plot title.
…: Additional arguments to persp if persp=TRUE, such as theta and phi.

Author

Edith Gabriel edith.gabriel@univ-avignon.fr

rinfec

Generate infection point patterns

Generate one (or several) realisation(s) of the infection process in a region \(S x T\).

Usage

rinfec(npoints, s.region, t.region, nsim=1, alpha, beta, gamma, 
s.distr="exponential", t.distr="uniform", maxrad, delta, h="step", 
g="min", recent=1, lambda=NULL, lmax=NULL, nx=100, ny=100, nt=1000, 
t0, inhibition=FALSE, ...)

Arguments

npoints: Number of points to simulate.
s.region: Two-column matrix specifying polygonal region containing all data locations. If s.region is missing, the unit square is considered.
t.region: Vector containing the minimum and maximum values of the time interval. If t.region is missing, the interval \([0,1]\) is considered.
nsim: Number of simulations to generate. Default is 1.
alpha: Numerical value for the latent period.
beta: Numerical value for the maximum infection rate.
gamma: Numerical value for the infection period.
h: Infection rate function which depends on alpha, beta and delta. Must be choosen among “step” and “gaussian”.
s.distr: Spatial distribution. Must be choosen among “uniform”, “gaussian”, “exponential” and “poisson”.
t.distr: Temporal distribution. Must be choosen among “uniform” and “exponential”.
maxrad: Single value or 2-vector of spatial and temporal maximum radiation respectively. If single value, the same value is used for space and time.
delta: Spatial distance of inhibition/contagion. If missing, the spatial radiation is used.
g: Compute the probability of acceptance of a new point from h and recent. Must be choosen among “min”, “max” and “prod”.
recent: If ``all consider all previous events. If is an integer, say \(N\), consider only the \(N\) most recent events.
lambda: Function or matrix defining the intensity of a Poisson process if s.distr is Poisson.
lmax: Upper bound for the value of lambda.
nx, ny: Define the 2-D grid on which the intensity is evaluated if s.distr is Poisson.
nt: Used to discretize time to compute the infection rate function.
t0: Minimum time used to compute the infection rate function. Default is the minimum of t.region.
inhibition: Logical. If TRUE, an inhibition process is generated. Otherwise, it is a contagious process.
…: Additional parameters if lambda is a function.

Value

xyt: Matrix (or list of matrices if nsim>1) containing the points \((x,y,t)\) of the simulated point pattern. xyt (or any element of the list if nsim>1) is an object of the class stpp.
s.region, t.region: Parameters passed in argument.

Examples

## Not run: 
# # inhibition; spatial distribution: uniform
# inf1 = rinfec(npoints=100, alpha=0.2, beta=0.6, gamma=0.5,
# maxrad=c(0.075,0.5), t.region=c(0,50), s.distr="uniform", 
# t.distr="uniform", h="gaussian", p="min", recent="all", t0=0.02, 
# inhibition=TRUE)
# animation(inf1$xyt, cex=0.8, runtime=10)
# 
# # contagion; spatial distribution: Poisson with intensity a given matrix
# data(fmd)
# data(northcumbria)
# h = mse2d(as.points(fmd[,1:2]), northcumbria, nsmse=30, range=3000)
# h = h$h[which.min(h$mse)]
# Ls = kernel2d(as.points(fmd[,1:2]), northcumbria, h, nx=50, ny=50)
# inf2 = rinfec(npoints=100, alpha=4, beta=0.6, gamma=20, maxrad=c(12000,20), 
# s.region=northcumbria, t.region=c(1,2000), s.distr="poisson", 
# t.distr="uniform", h="step", p="min", recent=1, 
# lambda=Ls$z, inhibition=FALSE)
# 
# 
# image(Ls$x, Ls$y, Ls$z, col=grey((1000:1)/1000)); polygon(northcumbria,lwd=2)
# animation(inf2$xyt, add=TRUE, cex=0.7, runtime=15)
# ## End(Not run)

Author

Edith Gabriel edith.gabriel@univ-avignon.fr, Peter J Diggle.

rinter

Generate interaction point patterns

Generate one (or several) realisation(s) of the inhibition or contagious process in a region \(S x T\).

Usage

rinter(npoints,s.region,t.region,hs="step",gs="min",thetas=0, deltas,ht="step",gt="min",thetat=1,deltat,recent="all",nsim=1, discrete.time=FALSE,replace=FALSE,inhibition=TRUE,...)

Arguments

npoints: Number of points to simulate.
s.region: Two-column matrix specifying polygonal region containing all data locations. If s.region is missing, the unit square is considered.
t.region: Vector containing the minimum and maximum values of the time interval. If t.region is missing, the interval \([0,1]\) is considered.
hs, ht: Function which depends on the distance between points and theta. Can be chosen among “step” and “gaussian” or can refer to a user defined function which only depend on d, theta, and delta (see details). If inhibition=TRUE, h is monotone, increasing, and must tend to 1 when the distance tends to infinity. 0 \(<=\)h(d,theta)\(<=\) 1. Otherwise, h is monotone, decreasing, and must tend to 1 when the distance tends to 0.
thetas, thetat: Parameters of hs and ht functions.
deltas, deltat: Spatial and temporal distance of inhibition.
gs, gt: Compute the probability of acceptance of a new point from hs or ht and recent. Must be choosen among “min”, “max” and “prod”.
recent: If ``all consider all previous events. If is an integer, say \(N\), consider only the \(N\) most recent events.
nsim: Number of simulations to generate. Default is 1.
discrete.time: If TRUE, times belong to \(N\), otherwise belong to \(R^+\).
replace: Logical. If TRUE allows times repeat.
inhibition: Logical. If TRUE, an inhibition process is generated. Otherwise, it is a contagious process.
…: Additional parameters if hs and ht are defined by the user.

Value

xyt: Matrix (or list of matrices if nsim>1) containing the points \((x,y,t)\) of the simulated point pattern. xyt (or any element of the list if nsim>1) is an object of the class stpp.
s.region, t.region: Parameters passed in argument.

Examples

# simple inhibition process
inh1 = rinter(npoints=200,thetas=0,deltas=0.05,thetat=0,deltat=0.001,
inhibition=TRUE)
## Not run: stan(inh1$xyt)

# inhibition process using hs and ht defined by the user
hs = function(d,theta,delta,mus=0.1)
{
 res=NULL
 a=(1-theta)/mus
 b=theta-a*delta
 for(i in 1:length(d))
    {   
    if (d[i]<=delta) res=c(res,theta)
    if (d[i]>(delta+mus)) res=c(res,1)
    if (d[i]>delta & d[i]<=(delta+mus)) res=c(res,a*d[i]+b)
    }
 return(res)
}
ht = function(d,theta,delta,mut=0.3)
{
 res=NULL
 a=(1-theta)/mut
 b=theta-a*delta
 for(i in 1:length(d))
    {   
    if (d[i]<=delta) res=c(res,theta)
    if (d[i]>(delta+mut)) res=c(res,1)
    if (d[i]>delta & d[i]<=(delta+mut)) res=c(res,a*d[i]+b)
    }
 return(res)
}
d=seq(0,1,length=100)
plot(d,hs(d,theta=0.2,delta=0.1,mus=0.1),xlab="",ylab="",type="l",
ylim=c(0,1),lwd=2,las=1)
lines(d,ht(d,theta=0.1,delta=0.05,mut=0.3),col=2,lwd=2)
legend("bottomright",col=1:2,lty=1,lwd=2,legend=c(expression(h[s]),
expression(h[t])),bty="n",cex=2)

inh2 = rinter(npoints=100, hs=hs, gs="min", thetas=0.2, deltas=0.1, 
ht=ht, gt="min", thetat=0.1, deltat=0.05, inhibition=TRUE)
## Not run: animation(inh2$xyt, runtime=15, cex=0.8)

# simple contagious process for given spatial and temporal regions
data(northcumbria)
cont1 = rinter(npoints=100, s.region=northcumbria, t.region=c(1,200), 
thetas=0, deltas=5000, thetat=0, deltat=10, recent=1, inhibition=FALSE)

## Not run: 
# plot(cont1$xyt,pch=19,s.region=cont1$s.region,mark=TRUE,mark.col=4)
# ## End(Not run)

Author

Edith Gabriel edith.gabriel@univ-avignon.fr, Peter J Diggle.

rlgcp

Generate log-Gaussian Cox point patterns

Generate one (or several) realisation(s) of the log-Gaussian cox process in a region \(S x T\).

Usage

rlgcp(s.region, t.region, replace=TRUE, npoints=NULL, nsim=1,  nx=100, ny=100, nt=100,separable=TRUE,model="exponential", param=c(1,1,1,1,1,1), scale=c(1,1),var.grf=1,mean.grf=0, lmax=NULL,discrete.time=FALSE,exact=FALSE,anisotropy=FALSE,ani.pars=NULL)

Arguments

s.region: Two-column matrix specifying polygonal region containing all data locations. If s.region is missing, the unit square is considered.
t.region: Vector containing the minimum and maximum values of the time interval. If t.region is missing, the interval \([0,1]\) is considered.
npoints: Number of points to simulate. If NULL, the number of points is from a Poisson distribution with mean the double integral of
\(\lambda(s,t)\) over s.region and t.region.
nsim: number of simulations to generate. Default is 1.
separable: Logical. If TRUE, the covariance function of the Gaussian random field is separable.
model: Vector of length 1 or 2 specifying the model(s) of covariance of the Gaussian random field. If separable=TRUE and model is of length 2, then the elements of model define the spatial and temporal covariances respectively. If separable=TRUE and model is of length 1, then the spatial and temporal covariances belongs to the same class of covariances, among “matern”, “exponential”, “stable”, “cauchy” and “wave” (see Details). If separable=FALSE, model must be of length 1 and is either “gneiting” or “cesare” (see Details).
param: \((\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5, \alpha_6)\). Vector of parameters of the covariance function (see Details).
scale: Vector of length 2 defining the spatial and temporal scale.
var.grf: Variance of the Gaussian random field.
mean.grf: Mean of the Gaussian random field.
replace: Logical allowing times repeat.
nx,ny,nt: Define the size of the 3-D grid on which the intensity is evaluated.
lmax: Upper bound for the value of \(\lambda(x,y,t)\).
discrete.time: If TRUE, times belong to \(N\), otherwise belong to \(R^+\).
exact: logical allowing exact simulation of Gaussian random fields (see manual for details).
anisotropy: If TRUE, simulate an anisotropic point pattern. Currently only implemented for separable covariance functions.
ani.pars: Vector of length 2, the anisotropy angle and the anisotropy ratio, respectively (see details).

Details

We implemented stationary, isotropic spatio-temporal covariance functions.

Separable covariance functions

\[c(h,t) = c_s(\| h \|) \, c_t(|t|) , h \in S, t \in T\]

The purely spatial and purely temporal covariance functions can be:

Exponential: \(c(r)=exp(-r)\),
Stable: \(c(r)=exp(-r^\alpha)\), \(\alpha in [0,2]\),
Cauchy: \(c(r)=(1+r^2)^(-\alpha)\), \(\alpha > 0\),
Wave: \(c(r)=sin(r)/r\) if \(r>0\), \(c(0)=1\),
Matern: \(c(r)={(\alpha r)^\nu}/{2^(\nu-1)\Gamma(\nu)} K_\nu(\alpha r)\), \(\nu>0\) and \(\alpha>0\).

\(K_\nu\) is the modified Bessel function of second kind: \[{\cal K}_{\nu}(x) = \frac{\pi}{2} \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin(\pi \nu)},\] with \(I_\nu(x) = (x/2)^\nu sum_{k=0}^{\infty} 1/(k! \Gamma(\nu+k+1)) (x/2)^(2k)\).

The parameters \(\alpha_1\) and \(\alpha_2\) correspond to the parameters of the spatial and temporal covariance respectively. For the Matern model, the parameters \(\alpha_1\), \(\alpha_3\) and \(\alpha_2\), \(\alpha_4\) correspond to the parameters \(\nu\), \(\alpha\) of the spatial and temporal covariance.

Non-separable covariance functions

The spatio-temporal covariance function can be:

Gneiting: \(c(h,t)=\psi(t^2/\beta_2)^(-\alpha_6) \phi( (h^2/\beta_1)/\psi(t^2/\beta_2) )\), \(\beta_1, \beta_2 >0\) are spatial and temporal scales respectively,
- If \(\alpha_2=1\), \(\phi(r)\) is the Stable model.
- if \(\alpha_2=2\), \(\phi(r)\) is the Cauchy model.
- If \(\alpha_2=3\), \(\phi(r)\) is the Matern model.
- If \(\alpha_5=1\), \(\psi(r) = (r^\alpha_3+1)^\alpha_4\),
- If \(\alpha_5=2\), \(\psi(r) = (\alpha_4^(-1) r^\alpha_3 +1)/(r^\alpha_3+1)\),
- If \(\alpha_5=3\), \(\psi(r) = log(r^\alpha_3+\alpha_4)/ log(\alpha_4)\),
The parameter \(\alpha_1\) is the respective parameter for the model of \(\phi(.)\), \(\alpha_3 in (0,1]\), \(\alpha_4 in (0,1]\) and \(\alpha_6 >= 2\).
cesare: \(c(h,t) = (1 + (h/\beta_1)^\alpha_1 + (t/\beta_2)^\alpha_2)^(-\alpha_3)\), \(\beta_1, \beta_2 >0\), \(\alpha_1, \alpha_2 in [1,2]\) and \(\alpha_3 >= 3/2\).

We also implemented anisotropic Log-Gaussian Cox processes. We considered geometric spatial anisotropy (see Moller and Toftaker, 2014). In this case the covariance function is elliptical and anisotropy is characterized by two parameters: the anisotropy angle \(\pi > \theta >=0\) and the anisotropy ratio \(1 >= \delta >0\) of the minor axis \(2 \omega \delta\) and the major axis \(2 \omega\).

\[C(h,t)=C_0\left( \sqrt{h \Sigma^{-1} h},t \right), \ h \in R^2.\]

Value

xyt: Matrix (or list of matrices if nsim>1) containing the points \((x,y,t)\) of the simulated point pattern. xyt (or any element of the list if nsim>1) is an object of the class stpp.
s.region, t.region: parameters passed in argument.
Lambda: \(nx x ny x nt\) array (or list of array if nsim>1) of the intensity.

References

Chan, G. and Wood A. (1997). An algorithm for simulating stationary Gaussian random fields. Applied Statistics, Algorithm Section, 46, 171–181.

Chan, G. and Wood A. (1999). Simulation of stationary Gaussian vector fields. Statistics and Computing, 9, 265–268.

Gneiting T. (2002). Nonseparable, stationary covariance functions for space-time data. Journal of the American Statistical Association, 97, 590–600.

Moller J. and Toftaker H. (2014). Geometric anisotropic spatial point pattern analysis and Cox processes. Scandinavian Journal of Statistics, 41, 414–435.

Examples

## Not run: 
# # non separable covariance function: 
# lgcp1 <- rlgcp(npoints=200, nx=50, ny=50, nt=50, separable=FALSE, 
# model="gneiting", param=c(1,1,1,1,1,2), var.grf=1, mean.grf=0)
# N <- lgcp1$Lambda[,,1];for(j in 2:(dim(lgcp1$Lambda)[3])){N <-
# N+lgcp1$Lambda[,,j]}
# image(N,col=grey((1000:1)/1000));box()
# animation(lgcp1$xyt, cex=0.8, runtime=10, add=TRUE, prevalent="orange")
# 
# # separable covariance function: 
# lgcp2 <- rlgcp(npoints=200, nx=50, ny=50, nt=50, separable=TRUE, 
# model="exponential", param=c(1,1,1,1,1,2), var.grf=2, mean.grf=-0.5*2)
# N <- lgcp2$Lambda[,,1];for(j in 2:(dim(lgcp2$Lambda)[3])){N <-
# N+lgcp2$Lambda[,,j]}
# image(N,col=grey((1000:1)/1000));box()
# animation(lgcp2$xyt, cex=0.8, pch=20, runtime=10, add=TRUE,
# prevalent="orange")
# ## End(Not run)

Author

Edith Gabriel edith.gabriel@univ-avignon.fr, Peter J Diggle.

rpcp

Generate Poisson cluster point patterns

Generate one (or several) realisation(s) of the Poisson cluster process in a region \(S x T\).

Usage

rpcp(s.region, t.region, nparents=NULL, npoints=NULL, lambda=NULL,  mc=NULL, nsim=1, cluster="uniform", dispersion, infectious=TRUE,  edge = "larger.region", larger.region=larger.region, tronc=1,...)

Arguments

s.region: Two-column matrix specifying polygonal region containing all data locations.
If s.region is missing, the unit square is considered.
t.region: Vector containing the minimum and maximum values of the time interval. If t.region is missing, the interval \([0,1]\) is considered.
nparents: Number of parents. If NULL, nparents is from a Poisson distribution with intensity lambda.
npoints: Number of points to simulate. If NULL (default), the number of points is from a Poisson distribution with mean the double integral of the intensity over s.region and t.region.
lambda: Intensity of the parent process. Can be either a numeric value, a function, or a 3d-array (see rpp). If NULL, it is constant and equal to nparents / volume of the domain.
mc: Average number of children per parent. It is used when npoints is NULL.
nsim: Number of simulations to generate.
cluster: Distribution of children: uniform,normal and ``exponential are currently implemented. Either a single value if the distribution in space and time is the same, or a vector of length 2, giving first the spatial distribution of children and then the temporal distribution.
dispersion: Scale parameter. It equals twice the standard deviation of location of children relative to their parent for a normal distribution of children; the mean for an exponential distribution and half range for an uniform distribution.
infectious: If TRUE, offsprings times are always greater than parents time.
edge: Specify the edge correction to use “larger.region” or “without”.
larger.region: By default, the larger spatial region is the convex hull of s.region enlarged by the spatial related value of dispersion and the larger time interval is t.region enlarged by the temporal related value of dispersion. One can over-ride default using the 2-vector parameter larger.region.
tronc: Parameter of the truncated exponential distribution for the distribution of children.
…: Additional parameters of the intensity of the parent process.

Value

xyt: Matrix (or list of matrices if nsim>1) containing the points \((x,y,t)\) of the simulated point pattern. xyt (or any element of the list if nsim>1) is an object of the class stpp.
s.region, t.region: Parameters passed in argument.

Examples

# homogeneous Poisson distribution of parents

data(northcumbria)
pcp1 <- rpcp(nparents=50, npoints=500, s.region=northcumbria, 
t.region=c(1,365), cluster=c("normal","exponential"), 
maxrad=c(5000,5))
## Not run: 
# animation(pcp1$xyt, s.region=pcp1$s.region, t.region=pcp1$t.region,
# runtime=5)
# ## End(Not run)
# inhomogeneous Poisson distribution of parents

lbda <- function(x,y,t,a){a*exp(-4*y) * exp(-2*t)}
pcp2 <- rpcp(nparents=50, npoints=500, cluster="normal", lambda=lbda, 
a=4000/((1-exp(-4))*(1-exp(-2))))
## Not run: 
# stan(pcp2$xyt)
# ## End(Not run)

Author

Edith Gabriel edith.gabriel@univ-avignon.fr, Peter J Diggle.

rpp

Generate Poisson point patterns

Generate one (or several) realisation(s) of the (homogeneous or inhomogeneous) Poisson process in a region \(S x T\).

Usage

rpp(lambda, s.region, t.region, npoints=NULL, nsim=1, replace=TRUE, discrete.time=FALSE, nx=100, ny=100, nt=100, lmax=NULL, ...)

Arguments

lambda: Spatio-temporal intensity of the Poisson process. If lambda is a single positive number, the function generates realisations of a homogeneous Poisson process, whilst if lambda is a function of the form \(lambda(x,y,t,...)\) or a 3D-array it generates realisations of an inhomogeneous Poisson process.
s.region: Two-column matrix specifying polygonal region containing all data locations. If s.region is missing, the unit square is considered.
t.region: Vector containing the minimum and maximum values of the time interval. If t.region is missing, the interval \([0,1]\) is considered.
replace: Logical allowing times repeat.
npoints: Number of points to simulate. If NULL, the number of points is from a Poisson distribution with mean the double integral of lambda over s.region and t.region.
discrete.time: If TRUE, times belong to \(N\), otherwise belong to \(R^+\).
nsim: Number of simulations to generate. Default is 1.
nx,ny,nt: Define the size of the 3-D grid on which the intensity is evaluated.
lmax: Upper bound for the value of \(lambda(x,y,t)\), if lambda is a function.
…: Additional parameters if lambda is a function.

Value

xyt: Matrix (or list of matrices if nsim>1) containing the points \((x,y,t)\) of the simulated point pattern. xyt (or any element of the list if nsim>1) is an object of the class stpp.
Lambda: \(nx x ny x nt\) array of the intensity surface at each time.
s.region, t.region, lambda: parameters passed in argument.

Examples

# Homogeneous Poisson process
# ---------------------------
hpp1 <- rpp(lambda=200,replace=FALSE)

## Not run: stan(hpp1$xyt)

# fixed number of points, discrete time, with time repeat.
data(northcumbria)
hpp2 <- rpp(npoints=500, s.region=northcumbria, t.region=c(1,1000), 
discrete.time=TRUE)
## Not run: 
# polymap(northcumbria)
# animation(hpp2$xyt, s.region=hpp2$s.region, t.region=hpp2$t.region, 
# runtime=10, add=TRUE)
# ## End(Not run)

## Not run: 
# # Inhomogeneous Poisson process
# # -----------------------------
# 
# # intensity defined by a function
# lbda1 = function(x,y,t,a){a*exp(-4*y) * exp(-2*t)}
# ipp1 = rpp(lambda=lbda1, npoints=400, a=3200/((1-exp(-4))*(1-exp(-2))))
# stan(ipp1$xyt)
# 
# # intensity defined by a matrix
# data(fmd)
# data(northcumbria)
# h = mse2d(as.points(fmd[,1:2]), northcumbria, nsmse=30, range=3000)
# h = h$h[which.min(h$mse)]
# Ls = kernel2d(as.points(fmd[,1:2]), northcumbria, h, nx=100, ny=100)
# Lt = dim(fmd)[1]*density(fmd[,3], n=200)$y
# Lst=array(0,dim=c(100,100,200))
# for(k in 1:200) Lst[,,k] <- Ls$z*Lt[k]/dim(fmd)[1]
# ipp2 = rpp(lambda=Lst, s.region=northcumbria, t.region=c(1,200), 
# discrete.time=TRUE)
#            
# par(mfrow=c(1,1))
# image(Ls$x, Ls$y, Ls$z, col=grey((1000:1)/1000)); polygon(northcumbria)
# animation(ipp2$xyt, add=TRUE, cex=0.5, runtime=15)
# ## End(Not run)

Author

Edith Gabriel edith.gabriel@univ-avignon.fr and Peter J Diggle.

sim.stpp

Generate spatio-temporal point patterns

Generate one (or several) realisation(s) of a spatio-temporal point process in a region \(S x T\).

Usage

sim.stpp(class="poisson", s.region, t.region, npoints=NULL, 
nsim=1, ...)

Arguments

class: Must be chosen among “poisson”, “cluster”, “cox”, “infectious” and “inhibition”.
s.region: Two-column matrix specifying polygonal region containing all data locations. If s.region is missing, the unit square is considered.
t.region: Vector containing the minimum and maximum values of the time interval. If t.region is missing, the interval \([0,1]\) is considered.
npoints: Number of points to simulate.
nsim: Number of simulations to generate. Default is 1.
…: Additional parameters related to the class parameter. See rpp for the Poisson process; rpcp for the Poisson cluster process; rlgcp for the Log-Gaussian Cox process; rinter for the interaction (inhibition or contagious) process and rinfec for the infectious process.

Value

xyt: Matrix (or list of matrices if nsim>1) containing the points \((x,y,t)\) of the simulated point pattern. xyt (or any element of the list if nsim>1) is an object of the class stpp.
s.region, t.region: Parameters passed in argument.

Author

Edith Gabriel edith.gabriel@univ-avignon.fr

stdcpp

Generate double-cluster point pattern

Generate a realisation of the double-cluster process in a region \(S x T\).

Usage

stdcpp(lambp, a, b, c, mu, s.region, t.region)

Arguments

s.region: Two-column matrix specifying polygonal region containing all data locations.If s.region is missing, the unit square is considered.
t.region: Vector containing the minimum and maximum values of the time interval. If t.region is missing, the interval \([0,1]\) is considered.
lambp: Intensity of the parent process. Can be either a numeric value, a function, or a 3d-array (see rpp).
a: Length of the semi-axes \(x\) of ellipsoid.
b: Length of the semi-axes \(y\) of ellipsoid.
c: Length of the semi-axes \(y\) of ellipsoid.
mu: Average number of daughter per parent. (a single positive number).

Details

We consider the straightforward extension of the classical Matern cluster process on the \(R^3\) case (with ellipsoid or balls) by considering the \(z\)-coordiantes as times.

Consider a Poisson point process in the plane with intensity \(\lambda_p\) as cluster centres for all times parent, as well as a ellipsoid (or ball) where the semi-axes are of lengths \(a\), \(b\) and \(c\), around of each Poisson point under a random general rotation. The scatter uniformly in all ellipsoid (or ball) of all points which are of the form \((x,y,z)\), the number of points in each cluster being random with a Poisson (\(\mu\)) distribution. The resulting point pattern is a spatio-temporal cluster point process with \(t=z\). This point process has intensity \(\lambda_{p} x \mu\).

Value

References

Baddeley, A., Rubak, E., Turner, R. (2015). Spatial Point Patterns: Methodology and Applications with R. CRC Press, Boca Raton.

Chiu, S. N., Stoyan, D., Kendall, W. S., and Mecke, J. (2013). Stochastic Geometry and its Applications. John Wiley & Sons.

Gabriel, E., Rowlingson, B., Diggle P J. (2013) stpp: an R package for plotting, simulating and analyzing Spatio-Temporal Point Patterns. Journal of Statistical Software 53, 1-29.

Illian, J B., Penttinen, A., Stoyan, H. and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. John Wiley and Sons, London.

Stoyan, D., Rodriguez-Cortes, F. J., Mateu, J., and Gille, W. (2017). Mark variograms for spatio-temporal point processes. Spatial Statistics. 20, 125-147.

Author

Francisco J. Rodriguez Cortes cortesf@uji.es

sthpcpp

Spatio-temporal hot-spots cluster point process model

Generate a realisation of the hot-spots cluster process in a region \(S x T\).

Usage

sthpcpp(lambp, r, mu, s.region, t.region)

Arguments

s.region: Two-column matrix specifying polygonal region containing all data locations.If s.region is missing, the unit square is considered.
t.region: Vector containing the minimum and maximum values of the time interval. If t.region is missing, the interval \([0,1]\) is considered.
lambp: Intensity of the Poisson process of cluster centres. A single positive number, a function, or a pixel image.
r: Radius parameter of the clusters.
mu: Average number of daughter per parent (a single positive number) or reference intensity for the cluster points (a function or a pixel image).

Details

This function generates a realisation of spatio-temporal cluster process, which can be considered as generalisation of the classical Matern cluster process, inside the spatio-temporal window.

Consider a Poisson point process in the plane with intensity \(\lambda_{p}\) as cluster centres for all times parent, as well as a infinite cylinder of radius \(R\) around of each Poisson point, orthogonal to the plane. The scatter uniformly in all cylinders of all points which are of the form \((x,y,z)\), the number of points in each cluster being random with a Poisson (\(\mu\)) distribution. The resulting point pattern is a spatio-temporal cluster point process with \(t=z\). This point process has intensity \(\lambda_{p} x \mu\).

Value

References

Baddeley, A., Rubak, E., Turner, R. (2015). Spatial Point Patterns: Methodology and Applications with R. CRC Press, Boca Raton.

Chiu, S. N., Stoyan, D., Kendall, W. S., and Mecke, J. (2013). Stochastic Geometry and its Applications. John Wiley & Sons.

Gabriel, E., Rowlingson, B., Diggle P J. (2013) stpp: an R package for plotting, simulating and analyzing Spatio-Temporal Point Patterns. Journal of Statistical Software 53, 1-29.

Illian, J B., Penttinen, A., Stoyan, H. and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. John Wiley and Sons, London.

Stoyan, D., Rodriguez-Cortes, F. J., Mateu, J., and Gille, W. (2017). Mark variograms for spatio-temporal point processes. Spatial Statistics. 20, 125-147.

Author

Francisco J. Rodriguez Cortes cortesf@uji.es

STIKhat

Estimation of the Space-Time Inhomogeneous K-function

Compute an estimate of the Space-Time Inhomogeneous K-function.

Usage

STIKhat(xyt, s.region, t.region, dist, times, lambda, 
correction="isotropic", infectious=FALSE)

Arguments

xyt: Coordinates and times \((x,y,t)\) of the point pattern.
s.region: Two-column matrix specifying polygonal region containing all data locations. If s.region is missing, the bounding box of xyt[,1:2] is considered.
t.region: Vector containing the minimum and maximum values of the time interval. If t.region is missing, the range of xyt[,3] is considered.
dist: Vector of distances \(u\) at which \(K(u,v)\) is computed. If missing, the maximum of dist is given by \(min(S_x, S_y)/4\), where \(S_x\) and \(S_y\) represent the maximum width and height of the bounding box of s.region.
times: Vector of times \(v\) at which \(K(u,v)\) is computed. If missing, the maximum of times is given by \((T_max - T_min)/4\), where \(T_min\) and \(T_max\) are the minimum and maximum of the time interval \(T\).
lambda: Vector of values of the space-time intensity function evaluated at the points \((x,y,t)\) in \(S x T\). If lambda is missing, the estimate of the space-time K-function is computed as for the homogeneous case (Diggle et al., 1995), i.e. considering \(n/|S x T|\) as an estimate of the space-time intensity.
correction: A character vector specifying the edge correction(s) to be applied among “isotropic”, “border”, “modified.border”, “translate” and “none” (see Details). The default is “isotropic”.
infectious: Logical value. If TRUE, only future events are considered and the isotropic edge correction method is used. See Details.

Details

Gabriel (2014) proposes the following unbiased estimator for the STIK-function, based on data giving the locations of events \(x_i: i = 1,...,n\) on a spatio-temporal region \(SxT\), where \(S\) is an arbitrary polygon and \(T\) is a time interval: \[\widehat{K}(u,v)=\sum_{i=1}^{n}\sum_{j\neq i}\frac{1}{w_{ij}}\frac{1}{\lambda(x_i)\lambda(x_j)}\mathbf{1}_{\lbrace \|s_i - s_j\| \leq u \ ; \ |t_i - t_j| \leq v \rbrace},\] where \(lambda(x_i)\) is the intensity at \(x_i = (s_i, t_i)\) and \(w_ij\) is an edge correction factor to deal with spatial-temporal edge effects. The edge correction methods implemented are:

translate: \(w_ij = |S intersect S_(s_i - s_j) x T intersect T_(t_i - t_j)|\), where \(S_(s_i - s_j)\) and \(T_(t_ i - t_j)\) are the translated spatial and temporal regions.

none: No edge correction is performed and \(w_ij = |S x T|\).

If parameter infectious = TRUE, ony future events are considered and the estimator is, using an isotropic edge correction factor (Gabriel and Diggle, 2009): \[\widehat{K}(u,v)=\frac{1}{|S\times T|}\frac{n}{n_v}\sum_{i=1}^{n_v}\sum_{j=1; j > i}^{n_v} \frac{1}{w_{ij}} \frac{1}{\lambda(x_i) \lambda(x_j)}\mathbf{1}_{\left\lbrace u_{ij} \leq u\right\rbrace}\mathbf{1}_{\left\lbrace t_j - t_i \leq v \right\rbrace}.\]

In this equation, the points \(x_i = (s_i, t_i)\) are ordered so that \(t_i < t_(i+1)\), with ties due to round-off error broken by randomly unrounding if necessary. To deal with temporal edge-effects, for each \(v\), \(n_v\) denotes the number of events for which \(t_i <= T_1 - v\), with \(T=[T_0, T_1]\). To deal with spatial edge-effects, we use Ripleys method.

If lambda is missing in argument, STIKhat computes an estimate of the space-time (homogeneous) K-function: \[\widehat{K}(u,v)=\frac{|S\times T|}{n_v(n-1)} \sum_{i=1}^{n_v}\sum_{j=1;j>i}^{n_v}\frac{1}{w_{ij}}\mathbf{1}_{\lbrace u_{ij}\leq u \rbrace}\mathbf{1}_{\lbrace t_j - t_i \leq v \rbrace}\]

Value

Khat: ndist x ntimes matrix containing values of \(K(u,v)\).
Ktheo: ndist x ntimes matrix containing theoretical values for a Poisson process; \(pi u^2 v\) for K and \(2 pi u^2 v\)) for K^*.
dist, times, infectious: Parameters passed in argument.
correction: The name(s) of the edge correction method(s) passed in argument.

References

Baddeley A., Moller J. and Waagepetersen R. (2000). Non- and semi-parametric estimation of interaction in inhomogeneous point patterns. Statistica Neerlandica, 54, 329–350.

Baddeley, A., Rubak, E., Turner, R., (2015). Spatial Point Patterns: Methodology and Applications with R. CRC Press, Boca Raton.

Diggle P. , Chedwynd A., Haggkvist R. and Morris S. (1995). Second-order analysis of space-time clustering. Statistical Methods in Medical Research, 4, 124–136.

Gabriel E., Diggle P. (2009). Second-order analysis of inhomogeneous spatio-temporal point process data. Statistica Neerlandica, 63, 43–51.

Gabriel E., Rowlingson B., Diggle P. (2013). stpp: an R package for plotting, simulating and analyzing Spatio-Temporal Point Patterns. Journal of Statistical Software, 53(2), 1–29.

Author

Edith Gabriel edith.gabriel@univ-avignon.fr

stpp-package: stpp-package, stpp

Space-Time Point Pattern simulation, visualisation and analysis

This package provides models of spatio-temporal point processes in a region \(S x T\) and statistical tools for analysing global and local second-order properties of such processes. It also includes static and dynamic (2D and 3D) plots. stpp is the first dedicated unified computational environment in the area of spatio-temporal point processes. The stpp package depends upon some other packages: splancs: spatial and space-time point pattern analysis rgl: interactive 3D plotting of densities and surfaces rpanel: simple interactive controls for R using tcltk package KernSmooth: functions for kernel smoothing for Wand & Jones (1995) plot3D: Tools for plotting 3-D and 2-D data

Details

stpp is a package for simulating, analysing and visualising patterns of points in space and time.

Following is a summary of the main functions and the dataset in the stpp package.

To visualise a spatio-temporal point pattern

animation: space-time data animation.
as.3dpoints: create data in spatio-temporal point format.
plot.stpp: plot spatio-temporal point object. Either a two-panels plot showing spatial locations and cumulative times, or a one-panel plot showing spatial locations with times treated as a quantitative mark attached to each location.
stan: 3D space-time animation.

To simulate spatio-temporal point patterns

rinfec: simulate an infection point process,
rinter: simulate an interaction (inhibition or contagious) point process,
rlgcp: simulate a log-Gaussian Cox point process,
rpcp: simulate a Poisson cluster point process,
rpp: simulate a Poisson point process,
stdcpp: simulate a double-cluster point process,
sthpcpp: simulate a hot-spot point process.

To analyse spatio-temporal point patterns

PCFhat: space-time inhomogeneous pair correlation function,
STIKhat: space-time inhomogeneous K-function,
ASTIKhat: Anisotropic space-time inhomogeneous K-function,
LISTAhat: space-time inhomogeneous pair correlation LISTA funcrions.

Dataset

fmd: 2001 food-and-mouth epidemic in north Cumbria (UK).

References

Baddeley A., Moller J. and Waagepetersen R. (2000). Non- and semi-parametric estimation of interaction in inhomogeneous point patterns. Statistica Neerlandica, 54, 329–350.

Chan, G. and Wood A. (1997). An algorithm for simulating stationary Gaussian random fields. Applied Statistics, Algorithm Section, 46, 171–181.

Chan, G. and Wood A. (1999). Simulation of stationary Gaussian vector fields. Statistics and Computing, 9, 265–268.

Diggle P. , Chedwynd A., Haggkvist R. and Morris S. (1995). Second-order analysis of space-time clustering. Statistical Methods in Medical Research, 4, 124–136.

Gabriel E. (2014). Estimating second-order characteristics of inhomogeneous spatio-temporal point processes: influence ofedge correction methods and intensity estimates. Methodology and computing in Applied Probabillity, 16(1).

Gabriel E., Diggle P. (2009). Second-order analysis of inhomogeneous spatio-temporal point process data. Statistica Neerlandica, 63, 43–51.

Gabriel E., Rowlingson B., Diggle P. (2013). stpp: an R package for plotting, simulating and analyzing Spatio-Temporal Point Patterns. Journal of Statistical Software, 53(2), 1–29.

Gneiting T. (2002). Nonseparable, stationary covariance functions for space-time data. Journal of the American Statistical Association, 97, 590–600.

Gonzalez, J. A., Rodriguez-Cortes, F. J., Cronie, O. and Mateu, J. (2016). Spatio-temporal point process statistics: a review. Spatial Statiscts, 18, 505–544.

Author

Edith Gabriel edith.gabriel@univ-avignon.fr, Barry Rowlingson, Peter J. Diggle and Francisco J. Rodriguez-Cortes

Space-Time Point Pattern Simulation, Visualisation and Analysis

Description

Depends

Suggests

Datasets available by data()

fmd

2001 food-and-mouth epidemic, north Cumbria (UK)

Usage

Format

References

See also

northcumbria

Polygon boundary of north Cumbria

Usage

Format

See also

High-Level Plots

stan

(3D) space-time data animation

Usage

Arguments

Details

Value

Author

Package Functions

animation

Space-time data animation

Usage

Arguments

Value

Author

as.3dpoint

Create data in spatio-temporal point format

Usage

Arguments

Value

Author

ASTIKhat

Anisotropic Space-Time Inhomogeneous \(K\)-function

Usage

Arguments

Value

References

Author

is.3dpoints

Spatio-temporal point objects

Usage

Arguments

Value

Author

LISTAhat

Estimation of the Space-Time Inhomogeneous Pair Correlation LISTA functions

Usage

Arguments

Details

Value

References

Author

PCFhat

Estimation of the Space-Time Inhomogeneous Pair Correlation function

Usage

Arguments

Details

Value

References

Author

plot.stpp: plot.stpp, plot

Plot for spatio-temporal point objects

Usage

Arguments

Value

References

See also

Author

plotK

Plot the estimation of the Space-Time Inhomogeneous K-function

Usage

Arguments

See also

Author