PCFhat: Estimation of the space-time pair correlation function

In stpp: Space-Time Point Pattern simulation, visualisation and analysis

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 = TRUE) 

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.
times	vector of times v at which K(u,v) is computed.
lambda	vector of values of the space-time intensity function evaluated at the points (x,y,t) in SxT. If lambda is missing, the estimate of the space-time pair correlation function is computed considering n/|SxT| 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	logical value. If TRUE, spatial (Ripley's) and temporal edge corrections are used.

Details

An approximately unbiased estimator for the space-time pair correlation function, based on data giving the locations of events xi: i=1...,n on a spatio-temporal region SxT, where S is an arbitrary polygon and T=[T0,T1]:

g(u,v) = 1/|SxT| sum_{i=1,...,n} sum_{j=1,...,n; j \neq j} 1/(wij*vij) ks(u - ||si-sj||)kt(v-|ti-tj|)/(lambda(xi)lambda(xj))

To deal with spatial edge-effects, we use Ripley's method, in which wij is the proportion of the circle centered on si and passing through sj, i.e. of radius uij=||si-sj||, that lies inside S. To deal with temporal edge effects, vij is equal to 1 if both ends of the interval of length 2|ti-tj| centred at ti lie within T and 1/2 otherwise.

ks() and kt() denotes kernel functions with bandwidth hs and ht. 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 g(u,v).
dist, times	parameters passed in argument.
kernel	a vector of names and bandwidths of the spatial and temporal kernels.

Author(s)

Edith Gabriel <[e-mail]>

References

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

Examples

## Not run: 
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=5000, ny=5000)
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
g <- PCFhat(xyt=FMD, dist=1:20, times=1:20, lambda=mhat*mut/dim(FMD)[1],
 s.region=northcumbria/1000,t.region=c(1,200))

# plotting the estimation plotPCF(g)
plotPCF(g,persp=TRUE,theta=-65,phi=35) 
## End(Not run)

stpp documentation built on May 2, 2019, 4:50 p.m.