Package 'stpp'

Space-time data animation

Description

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(s)

Peter J Diggle, Edith Gabriel <[email protected]>.

---

Create data in spatio-temporal point format

Description

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(s)

Edith Gabriel <[email protected]>, Peter Diggle, Barry Rowlingson.

---

Anisotropic Space-Time Inhomogeneous $K$-function

Description

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 $\widehat{K}_{\phi}(r,t)$ 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 $\widehat{K}_{\phi}(r,t)$ 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 $S\times T$. If lambda is missing, the estimate of the anisotropic space-time $K$-function is computed as for the homogeneous case, i.e. considering $n/|S\times T|$ as an estimate of the space-time intensity.
ang	Angle in radians at which $\widehat{K}_{\phi}(r,t)$ 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

A list containing:

AKhat	ndist x ntimes matrix containing values of $\widehat{K}_{\phi}(u,t)$.
dist, times	Parameters passed in argument.
correction	The name(s) of the edge correction method(s) passed in argument.

Author(s)

Francisco J. Rodriguez-Cortes <[email protected]>

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.

---

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

Description

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.

See Also

northcumbria for boundaries of the county of north Cumbria.

---

Spatial mark variogram function

Description

Computes an estimator of the spatial mark variogram function.

Usage

gsp(xyt,s.region,s.lambda,ds,ks="epanech",hs,correction="none",approach="simplified")

Arguments

xyt	Spatial coordinates and times $(x,y,t)$ of the point pattern.
s.region	Two-column matrix specifying polygonal region containing all data locations.
s.lambda	Vector of values of the spatial intensity function evaluated at the points $(x,y)$ in $W$. If s.lambda is missing, the estimate of the spatial mark correlation function is computed as for the homogeneous case, i.e. considering $n/|W|$ as an estimate of the spatial intensity under the parameter approach="standardised".
ds	A vector of distances u at which gsp(u) is computed.
ks	A kernel function for the spatial distances. The default is the "epanech" kernel. It can also be "box" for the uniform kernel, or "biweight".
hs	A bandwidth of the kernel function ks.
correction	A character vector specifying the edge-correction(s) to be applied among "isotropic", "border", "modified.border", "translate", "setcovf" and "none". The default is "none".
approach	A character vector specifying the approach to use for the estimation to be applied among "simplified" or "standardised". If approach is missing, "simplified" is considered by default.

Details

By default, this command calculates an estimate of the spatial mark variogram function $\gamma_[sp](r)$ for a spatio-temporal point pattern.

Value

egsp	A vector containing the values of $\gamma_{sp}(r)$ estimated
ds	If ds is missing, a vector of distances u at which gsp(u) is computed from 0 to until quarter of the maximum distance between the points in the pattern.
kernel	A vector of names and bandwidth of the spatial kernel.
gsptheo	Value under the Poisson case is calculated considering $\tau$=max(xyt[,3])-min(xyt[,3]).

Author(s)

Francisco J. Rodriguez Cortes <[email protected]> https://fjrodriguezcortes.wordpress.com

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.

Examples

## Not run:
#################

# A realisation of spatio-temporal homogeneous Poisson point processes
hpp <- rpp(lambda = 100, replace = FALSE)$xyt

# R plot
plot(hpp)

# This function provides an kernel estimator of the spatial mark variogram function
out <- gsp(hpp)

# R plot - Spatial mark variogram function
par(mfrow=c(1,1))
xl <- c(0,0.25)
yl <- c(0,max(out$gsptheo,out$egsp))
plot(out$ds,out$egsp,type="l",xlab="r = distance",ylab=expression(gamma[sp](r)),
                 xlim=xl,ylim=yl,col=1,cex.lab=1.5,cex.axis=1.5)
lines(out$ds,rep(out$gsptheo,length(out$ds)),col=11)

## End(Not run)

---

Temporal mark variogram function

Description

Computes an estimator of the temporal mark variogram function.

Usage

gte(xyt,t.region,t.lambda,dt,kt="epanech",ht,correction="none",approach="simplified")

Arguments

xyt	Spatial coordinates and times $(x,y,t)$ of the point pattern.
t.region	Vector containing the minimum and maximum values of the time interval.
t.lambda	Vector of values of the temporal intensity function evaluated at the points $t$ in $T$. If t.lambda is missing, the estimate of the temporal mark correlation function is computed as for the homogeneous case, i.e. considering $n/|T|$ as an estimate of the temporal intensity under the parameter approach="standardised".
dt	A vector of times v at which gte(v) is computed.
kt	A kernel function for the temporal distances. The default is the "epanech" kernel. It can also be "box" for the uniform kernel, or "biweight".
ht	A bandwidth of the kernel function kt.
correction	A character vector specifying the edge-correction(s) to be applied among "isotropic", "border", "modified.border", "translate", "setcovf" and "none". The default is "none".
approach	A character vector specifying the approach to use for the estimation to be applied among "simplified" or "standardised". If approach is missing, "simplified" is considered by default.

Details

By default, this command calculates an estimate of the temporal mark variogram function $\gamma_[te](t)$ for a spatio-temporal point pattern.

Value

egte	A vector containing the values of $\gamma_[te](v)$ estimated.
dt	Parameter passed in argument. If dt is missing, a vector of temporal distances v at which gte(v) is computed from 0 to until quarter of the maximum distance between the times in the temporal pattern.
kernel	Parameters passed in argument. A vector of names and bandwidth of the spatial kernel.
gtetheo	Value under the Poisson case is calculated considering the side lengths of the bounding box of xyt[,1:2].

Author(s)

Francisco J. Rodriguez Cortes <[email protected]> https://fjrodriguezcortes.wordpress.com

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.

Examples

## Not run:
#################

# A realisation of spatio-temporal homogeneous Poisson point processes
hpp <- rpp(lambda = 100, replace = FALSE)$xyt

# R plot
plot(hpp)

# This function provides an kernel estimator of the temporal mark variograma function
out <- gte(hpp)

# R plot - Temporal mark variogram function
par(mfrow=c(1,1))
xl <- c(0,0.25)
yl <- c(0,max(out$egte,out$gtetheo))
plot(out$dt,out$egte,type="l",xlab="t = time",ylab=expression(gamma[te](t)),
                 xlim=xl,ylim=yl,col=1,cex.lab=1.5,cex.axis=1.5)
lines(out$dt,rep(out$gtetheo,length(out$dt)),col=11)

## End(Not run)

---

Spatio-temporal point objects

Description

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(s)

Edith Gabriel <[email protected]>, Peter Diggle, Barry Rowlingson.

---

Estimation of the Space-Time Inhomogeneous K LISTA functions

Description

Compute an estimate of the space-time K LISTA functions.

Usage

KLISTAhat(xyt, s.region, t.region, dist, times, lambda, 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 $K^{(i)}(u,v)$ 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 $K^{(i)}(u,v)$ 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 $S\times T$. If lambda is missing, the estimate of the space-time pair correlation function is computed as for the homogeneous case, i.e. considering $(n-1)/|S \times T|$ 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 PCFhat). The default is "isotropic".

Details

An individual product density LISTA functions $K^{(i)}(.,.)$ should reveal the extent of the contribution of the event $(u_i,t_i)$ to the global estimator of the K-function $K(.,.)$, 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. (2019).

Value

A list containing:

list.KLISTA	A list containing the values of the estimation of $K^{(i)}(r,t)$ for each one of $n$ points of the point pattern by matrixs.
klistatheo	ndist x ntimes matrix containing theoretical values for a Poisson process.
dist, times	Parameters passed in argument.
correction	The name(s) of the edge correction method(s) passed in argument.

Author(s)

Francisco J. Rodriguez-Cortes <[email protected]>

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., Adelfio, G., Mateu, J. and Rodriguez-Cortes, F. J. (2019). Some properties of weighted local second-order statistcs for spatio-temporal point process. Submitted.

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

---

Spatial mark correlation function

Description

Computes an estimator of the spatial mark correlation function.

Usage

kmmr(xyt,s.region,s.lambda,ds,ks="epanech",hs,correction="none",approach="simplified")

Arguments

xyt	Spatial coordinates and times $(x,y,t)$ of the point pattern.
s.region	Two-column matrix specifying polygonal region containing all data locations.
s.lambda	Vector of values of the spatial intensity function evaluated at the points $(x,y)$ in $W$. If s.lambda is missing, the estimate of the spatial mark correlation function is computed as for the homogeneous case, i.e. considering $n/|W|$ as an estimate of the spatial intensity under the parameter approach="standardised".
ds	A vector of distances u at which kmmr(u) is computed.
ks	A kernel function for the spatial distances. The default is the "epanech" kernel. It can also be "box" for the uniform kernel, or "biweight".
hs	A bandwidth of the kernel function ks.
correction	A character vector specifying the edge-correction(s) to be applied among "isotropic", "border", "modified.border", "translate", "setcovf" and "none". The default is "none".
approach	A character vector specifying the approach to use for the estimation to be applied among "simplified" or "standardised". If approach is missing, "simplified" is considered by default.

Details

By default, this command calculates an estimate of the spatial mark correlation function $k_[mm](r)$ for a spatio-temporal point pattern.

Value

ekmmr	A vector containing the values of $k_[mm](u)$ estimated.
ds	If ds is missing, a vector of distances u at which kmmr(u) is computed from 0 to until quarter of the maximum distance between the points in the pattern.
kernel	A vector of names and bandwidth of the spatial kernel.
kmmrtheo	Value under the Poisson case is calculated considering $\tau$=max(xyt[,3])-min(xyt[,3]).

Author(s)

Francisco J. Rodriguez Cortes <[email protected]> https://fjrodriguezcortes.wordpress.com

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 Wilfried, G. (2016). Mark variograms for spatio-temporal point processes, Submitted .

Examples

## Not run:
    #################
    
    # A realisation of spatio-temporal homogeneous Poisson point processes
    hpp <- rpp(lambda = 100, replace = FALSE)$xyt
    
    # R plot
    plot(hpp)
    
    # This function provides an kernel estimator of the spatial mark correlation function
    out <- kmmr(hpp)
    
    # R plot - Spatial mark correlation function
    par(mfrow=c(1,1))
    xl <- c(0,0.25)
    plot(out$ds,out$ekmmr,type="l",xlab="r = distance",ylab=expression(k[mm](r)),
         xlim=xl,col=1,cex.lab=1.5,cex.axis=1.5)
    lines(out$ds,rep(out$kmmrtheo,length(out$ds)),col=11)
    
    ## End(Not run)

---

Temporal mark correlation function

Description

Computes an estimator of the temporal mark correlation function.

Usage

kmmt(xyt,t.region,t.lambda,dt,kt="epanech",ht,correction="none",approach="simplified")

Arguments

xyt	Spatial coordinates and times $(x,y,t)$ of the point pattern.
t.region	Vector containing the minimum and maximum values of the time interval.
t.lambda	Vector of values of the temporal intensity function evaluated at the points $t$ in $T$. If t.lambda is missing, the estimate of the temporal mark correlation function is computed as for the homogeneous case, i.e. considering $n/|T|$ as an estimate of the temporal intensity under the parameter approach="standardised".
dt	A vector of times v at which kmmt(v) is computed.
kt	A kernel function for the temporal distances. The default is the "epanech" kernel. It can also be "box" for the uniform kernel, or "biweight".
ht	A bandwidth of the kernel function kt.
correction	A character vector specifying the edge-correction(s) to be applied among "isotropic", "border", "modified.border", "translate", "setcovf" and "none". The default is "none".
approach	A character vector specifying the approach to use for the estimation to be applied among "simplified" or "standardised". If approach is missing, "simplified" is considered by default.

Details

By default, this command calculates an estimate of the temporal mark correlation function $k_[mm](t)$ for a spatio-temporal point pattern.

Value

ekmmt	A vector containing the values of $k_[mm](v)$ estimated.
dt	Parameter passed in argument. If dt is missing, a vector of temporal distances v at which kmmt(v) is computed from 0 to until quarter of the maximum distance between the times in the temporal pattern.
kernel	Parameters passed in argument. A vector of names and bandwidth of the spatial kernel.
kmmttheo	Value under the Poisson case is calculated considering the side lengths of the bounding box of xyt[,1:2].

Author(s)

Francisco J. Rodriguez Cortes <[email protected]> https://fjrodriguezcortes.wordpress.com

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 Wilfried, G. (2016). Mark variograms for spatio-temporal point processes, Submitted .

Examples

## Not run:
    #################
    
    # A realisation of spatio-temporal homogeneous Poisson point processes
    hpp <- rpp(lambda = 100, replace = FALSE)$xyt
    
    # R plot
    plot(hpp)
    
    # This function provides an kernel estimator of the temporal mark correlation function
    out <- kmmt(hpp)
    
    # R plot - Temporal mark correlation function
    par(mfrow=c(1,1))
    xl <- c(0,0.25)
    plot(out$dt,out$ekmmt,type="l",xlab="t = time",ylab=expression(k[mm](t)),
         xlim=xl,col=1,cex.lab=1.5,cex.axis=1.5)
    lines(out$dt,rep(out$kmmttheo,length(out$dt)),col=11)
    
    ## End(Not run)

---

Spatial $r$-mark function

Description

Computes an estimator of the spatial $r$-mark function.

Usage

kmr(xyt,s.region,s.lambda,ds,ks="epanech",hs,correction="none",approach="simplified")

Arguments

xyt	Spatial coordinates and times $(x,y,t)$ of the point pattern.
s.region	Two-column matrix specifying polygonal region containing all data locations.
s.lambda	Vector of values of the spatial intensity function evaluated at the points $(x,y)$ in $W$. If s.lambda is missing, the estimate of the spatial mark correlation function is computed as for the homogeneous case, i.e. considering $n/|W|$ as an estimate of the spatial intensity under the parameter approach="standardised".
ds	A vector of distances u at which kmr(u) is computed.
ks	A kernel function for the spatial distances. The default is the "epanech" kernel. It can also be "box" for the uniform kernel, or "biweight".
hs	A bandwidth of the kernel function ks.
correction	A character vector specifying the edge-correction(s) to be applied among "isotropic", "border", "modified.border", "translate", "setcovf" and "none". The default is "none".
approach	A character vector specifying the approach to use for the estimation to be applied among "simplified" or "standardised". If approach is missing, "simplified" is considered by default.

Details

By default, this command calculates an estimate of the spatial $r$-mark function $k_[m.](r)$ for a spatio-temporal point pattern.

Value

ekmr	A vector containing the values of $k_[m.](u)$ estimated.
ds	If ds is missing, a vector of distances u at which kmr(u) is computed from 0 to until quarter of the maximum distance between the points in the pattern.
kernel	A vector of names and bandwidth of the spatial kernel.
kmrtheo	Value under the Poisson case is calculated considering $\tau$=max(xyt[,3])-min(xyt[,3]).

Author(s)

Francisco J. Rodriguez Cortes <[email protected]> https://fjrodriguezcortes.wordpress.com

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 Wilfried, G. (2016). Mark variograms for spatio-temporal point processes, Submitted .

Examples

## Not run:
    #################
    
    # A realisation of spatio-temporal homogeneous Poisson point processes
    hpp <- rpp(lambda = 100, replace = FALSE)$xyt
    
    # R plot
    plot(hpp)
    
    # This function provides an kernel estimator of the spatial r-mark function
    out <- kmr(hpp)
    
    # R plot - Spatial r-mark function
    par(mfrow=c(1,1))
    xl <- c(0,0.25)
    plot(out$ds,out$ekmr,type="l",xlab="r = distance",ylab=expression(k[m.](r)),
         xlim=xl,col=1,cex.lab=1.5,cex.axis=1.5)
    lines(out$ds,rep(out$kmrtheo,length(out$ds)),col=11)
    
    ## End(Not run)

---

Temporal $t$-mark function

Description

Computes an estimator of the temporal $t$-mark function.

Usage

kmt(xyt,t.region,t.lambda,dt,kt="epanech",ht,correction="none",approach="simplified")

Arguments

xyt	Spatial coordinates and times $(x,y,t)$ of the point pattern.
t.region	Vector containing the minimum and maximum values of the time interval.
t.lambda	Vector of values of the temporal intensity function evaluated at the points $t$ in $T$. If t.lambda is missing, the estimate of the temporal mark correlation function is computed as for the homogeneous case, i.e. considering $n/|T|$ as an estimate of the temporal intensity under the parameter approach="standardised".
dt	A vector of times v at which kmt(v) is computed.
kt	A kernel function for the temporal distances. The default is the "epanech" kernel. It can also be "box" for the uniform kernel, or "biweight".
ht	A bandwidth of the kernel function kt.
correction	A character vector specifying the edge-correction(s) to be applied among "isotropic", "border", "modified.border", "translate", "setcovf" and "none". The default is "none".
approach	A character vector specifying the approach to use for the estimation to be applied among "simplified" or "standardised". If approach is missing, "simplified" is considered by default.

Details

By default, this command calculates an estimate of the temporal $t$-mark function $k_[m.](t)$ for a spatio-temporal point pattern.

Value

ekmt	A vector containing the values of $k_[m.](v)$ estimated.
dt	Parameter passed in argument. If dt is missing, a vector of temporal distances v at which kmt(v) is computed from 0 to until quarter of the maximum distance between the times in the temporal pattern.
kernel	Parameters passed in argument. A vector of names and bandwidth of the spatial kernel.
kmttheo	Value under the Poisson case is calculated considering the side lengths of the bounding box of xyt[,1:2].

Author(s)

Francisco J. Rodriguez Cortes <[email protected]> https://fjrodriguezcortes.wordpress.com

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 Wilfried, G. (2016). Mark variograms for spatio-temporal point processes, Submitted .

Examples

## Not run:
    #################
    
    # A realisation of spatio-temporal homogeneous Poisson point processes
    hpp <- rpp(lambda = 100, replace = FALSE)$xyt
    
    # R plot
    plot(hpp)
    
    # This function provides an kernel estimator of the temporal t-mark function
    out <- kmt(hpp)
    
    # R plot - Temporal t-mark function
    par(mfrow=c(1,1))
    xl <- c(0,0.25)
    plot(out$dt,out$ekmt,type="l",xlab="t = time",ylab=expression(k[m.](t)),
         xlim=xl,col=1,cex.lab=1.5,cex.axis=1.5)
    lines(out$dt,rep(out$kmttheo,length(out$dt)),col=11)
    
    ## End(Not run)

---

Estimation of the Space-Time Inhomogeneous Pair Correlation LISTA functions

Description

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 $S\times T$. If lambda is missing, the estimate of the space-time pair correlation function is computed as for the homogeneous case, i.e. considering $(n-1)/|S \times T|$ 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

A list containing:

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.

Author(s)

Francisco J. Rodriguez-Cortes <[email protected]>

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.

---

Polygon boundary of north Cumbria

Description

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.

See Also

fmd for the space-time pattern of food-and-mouth disease in this county in 2001.

---

Estimation of the Space-Time Inhomogeneous Pair Correlation function

Description

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 $S\times T$. If lambda is missing, the estimate of the space-time pair correlation function is computed as for the homogeneous case, i.e. considering $n/|S \times T|$ 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 $S \times T$, 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: $w_{ij} = |S \times T| w_{ij}^{(t)} w_{ij}^{(s)}$, 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 Ripley's edge correction factor).

border: $w_{ij}=\frac{\sum_{j=1}^{n}\mathbf{1}\lbrace d(s_j,S)>u \ ; \ d(t_j,T) >v\rbrace/\lambda(x_j)}{\mathbf{1}_{\lbrace d(s_i,S) > u \ ; \ d(t_i,T) >v \rbrace}}$, 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: $w_{ij} = \frac{|S_{\ominus u}|\times|T_{\ominus v}|}{\mathbf{1}_{\lbrace d(s_i,S) > u \ ; \ d(t_i,T) >v \rbrace}}$, where $S_{\ominus u}$ and $T_{\ominus v}$ 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: $w_{ij} =|S \cap S_{s_i-s_j}| \times |T \cap T_{t_i-t_j}|$, 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 $w_{ij}=|S \times T|$.

$k_s()$ 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

A list containing:

pcf	ndist x ntimes matrix containing values of $\hat{g}(u,v)$.
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.

Author(s)

Edith Gabriel <[email protected]>

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.

Examples

# First example

data(fmd)
data(northcumbria)
FMD<-as.3dpoints(fmd[,1]/1000,fmd[,2]/1000,fmd[,3])
Northcumbria=northcumbria/1000

# estimation of the temporal intensity
Mt<-density(FMD[,3],n=1000)
mut<-Mt$y[findInterval(FMD[,3],Mt$x)]*dim(FMD)[1]

# estimation of the spatial intensity
h<-mse2d(as.points(FMD[,1:2]), Northcumbria, nsmse=50, range=4)
h<-h$h[which.min(h$mse)]
Ms<-kernel2d(as.points(FMD[,1:2]), Northcumbria, h, nx=500, ny=500)
atx<-findInterval(x=FMD[,1],vec=Ms$x)
aty<-findInterval(x=FMD[,2],vec=Ms$y)
mhat<-NULL
for(i in 1:length(atx)) mhat<-c(mhat,Ms$z[atx[i],aty[i]])

# estimation of the pair correlation function
g1 <- PCFhat(xyt=FMD, dist=1:15, times=1:15, lambda=mhat*mut/dim(FMD)[1],
 s.region=northcumbria/1000,t.region=c(1,200))

# plotting the estimation 

plotPCF(g1)
plotPCF(g1,type="persp",theta=-65,phi=35) 


# Second example

xyt=rpp(lambda=200)
g2=PCFhat(xyt$xyt,dist=seq(0,0.16,by=0.02),
times=seq(0,0.16,by=0.02),correction=c("border","translate"))

plotPCF(g2,type="contour",which="border")

---

Plot for spatio-temporal point objects

Description

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

## S3 method for class 'stpp'
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 style="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

None

Author(s)

Edith Gabriel <[email protected]> and Francisco J. Rodriguez-Cortes.

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.

See Also

as.3dpoints for creating data in spatio-temporal point format.

---

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

Description

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 $K_{ST}(u,v)$ or $L(u,v)=K_{ST}(u,v)-\pi u^2 v$ 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(s)

Edith Gabriel <[email protected]>

See Also

contour, persp, image and STIKhat for an example.

---

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

Description

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(s)

Edith Gabriel <[email protected]>

See Also

contour, persp, image and PCFhat for an example.

---

Generate infection point patterns

Description

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

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

A list containing:

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(s)

Edith Gabriel <[email protected]>, Peter J Diggle.

See Also

plot.stpp, animation and stan for plotting space-time point patterns.

Examples

# 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)
plot(inf1$xyt, style="elegant")


# 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)

---

Generate interaction point patterns

Description

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

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 $\leq$h(d,theta)$\leq$ 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 ${\bf R}^+$.
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

A list containing:

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(s)

Edith Gabriel <[email protected]>, Peter J Diggle.

See Also

plot.stpp, animation and stan for plotting space-time point patterns.

Examples

# simple inhibition process
inh1 = rinter(npoints=200,thetas=0,deltas=0.05,thetat=0,deltat=0.001,
inhibition=TRUE)
plot(inh1$xyt,style="elegant")


# 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)
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=10000, thetat=0, deltat=10, recent=1, inhibition=FALSE)
plot(cont1$xyt,pch=19,s.region=cont1$s.region,mark=TRUE,mark.col=4)

---

Generate log-Gaussian Cox point patterns

Description

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

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: $c(r) = \frac{\sin(r)}{r}$ if $r>0$, $c(0)=1$,

• Matern: $c(r) = \frac{(\alpha r)^\nu}{2^{\nu-1}\Gamma(\nu)} {\cal K}_{\nu}(\alpha r)$, $\nu > 0$ and $\alpha > 0$.

  ${\cal K}_{\nu}$ 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 $I_{\nu}(x) = \left( \frac{x}{2} \right)^{\nu} \sum_{k=0}^\infty \frac{1}{k! \Gamma(\nu+k+1)} \left( \frac{x}{2} \right)^{2k}$.

  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: $c(h,t) = \psi (t^2/\beta_2)^{-\alpha_6} \phi \left(\frac{h^2/ \beta_1}{\psi (t^2/\beta_2)} \right)$, $\beta_1, \beta_2 >0$ 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 $\phi(\cdot)$, $\alpha_3 \in (0,1]$, $\alpha_4 \in (0,1]$ and $\alpha_6 \geq 2$.

• cesare: $c(h,t) = \left( 1 + (h/\beta_1)^{\alpha_1} + (t/\beta_2)^{\alpha_2} \right)^{-\alpha_3}$, $\beta_1, \beta_2 >0$, $\alpha_1, \alpha_2 \in [1,2]$ and $\alpha_3 \geq 3/2$.

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 $0 \leq \theta < \pi$ and the anisotropy ratio $0 < \delta \leq 1$ 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

A list containing:

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	$nx \times ny \times nt$ array (or list of array if nsim>1) of the intensity.

Author(s)

Edith Gabriel <[email protected]>, Peter J Diggle.

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.

See Also

plot.stpp, animation and stan for plotting space-time point patterns.

Examples

# 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")

# anisotropic

sigma2=0.5
simlgcp <- rlgcp(npoints=500,nx=250, ny=200, nt=50,separable=TRUE,
s.region=matrix(c(0,2,2,0,0,0,0.5,0.5),byrow=FALSE,ncol=2), model="exponential", 
param=c(1,1,1,1,1,2), var.grf=sigma2, mean.grf=-0.5*sigma2,anisotropy = TRUE,
ani.pars = c(pi/4,0.1))

N <- simlgcp$Lambda[,,1];for(j in 2:dim(simlgcp$Lambda)[3]){N <- N+simlgcp$Lambda[,,j]}
image(x=simlgcp$grid[[1]]$x,y=simlgcp$grid[[1]]$y,z=N,col=grey((1000:1)/1000));box()
points(simlgcp$xyt[,1:2],pch=19,cex=0.25,col=2)

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Generate Poisson cluster point patterns

Description

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

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, offspring's times are always greater than parent's 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

A list containing:

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(s)

Edith Gabriel <[email protected]>, Peter J Diggle.

See Also

plot.stpp, animation and stan for plotting space-time point patterns.

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))

animation(pcp1$xyt, s.region=pcp1$s.region, t.region=pcp1$t.region,
runtime=5)

# 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))))
plot(pcp2$xyt, style="elegant")

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