Description Usage Arguments Details Value Author(s) References Examples
Compute an estimate of the spacetime pair correlation function.
1 2 
xyt 
Coordinates and times (x,y,t) of the point pattern. 
s.region 
Twocolumn matrix specifying polygonal region containing all data locations. If 
t.region 
Vector containing the minimum and maximum values of the time interval. If 
dist 
Vector of distances u at which g(u,v) is computed. If missing, the maximum of 
times 
Vector of times v at which g(u,v) is computed. If missing, the maximum of 
lambda 
Vector of values of the spacetime intensity function evaluated at the points (x,y,t) in S x T. If 
ks 
Kernel function for the spatial distances. Default is the 
hs 
Bandwidth of the kernel function 
kt 
Kernel function for the temporal distances. Default is the 
ht 
Bandwidth of the kernel function 
correction 
A character vector specifying the edge correction(s) to be applied among 
An approximately unbiased estimator for the spacetime pair correlation function, based on data giving the locations of events x_i: i = 1,...,n on a spatiotemporal region SxT, where S is an arbitrary polygon and T a time interval:
g(u,v) = 1/SxT 1/(4 pi u) sum_{i = 1,...,n} sum_{j = 1,...,n; j != i} 1/w_ij 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 spatialtemporal 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 2t_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 = (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.
A list containing:
pcf 

pcftheo 

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. 
Edith Gabriel <[email protected]>
Baddeley, A., Rubak, E., Turner, R., (2015). Spatial Point Patterns: Methodology and Applications with R. CRC Press, Boca Raton.
Gabriel E., Diggle P. (2009). Secondorder analysis of inhomogeneous spatiotemporal point process data. Statistica Neerlandica, 63, 43–51.
Gabriel E., Rowlingson B., Diggle P. (2013). stpp: an R package for plotting, simulating and analyzing SpatioTemporal Point Patterns. Journal of Statistical Software, 53(2), 1–29.
Gabriel E. (2014). Estimating secondorder characteristics of inhomogeneous spatiotemporal 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.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37  # 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")

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