STIKhat: Estimation of the Space-Time Inhomogeneous K-function
In stpp: Space-Time Point Pattern simulation, visualisation and analysis

Description Usage Arguments Details Value Author(s) References Examples

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

1 2	STIKhat(xyt, s.region, t.region, dist, times, lambda, correction=TRUE, infectious=TRUE)

`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 K-function is computed as for the homogeneous case (Diggle et al., 1995), i.e. considering n/\|SxT\| as an estimate of the space-time intensity.
`correction`	logical value. If `TRUE`, spatial (Ripley's) and temporal edge corrections are used.
`infectious`	logical value. If `TRUE`, only future events are considered. See Details.

Gabriel and Diggle (2009) propose the following approximately unbiased estimator for the STIK-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]:

K(u,v) = 1/|SxT| n/nv sum_{i=1,...,nv} sum_{j=1,...,nv; j > i} 1/wij 1/(lambda(x_i)lambda(x_j)) 1{uij <= u} 1{tj - ti <= v}

In this equation, lambda(xi) is the intensity at xi=(si,ti) and the xi are ordered so that ti < 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, nv denotes the number of events for which ti <= T1-v. 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.

If lambda is missing in argument, STIKhat computes an estimate of the space-time (homogeneous) K-function:

K'(u,v) = |SxT|/(nv(n-1)) sum_{i=1,...,nv} sum_{j=1,...,nv; j>i} 1/wij 1{uij <= u} 1{tj - ti <= v}

If parameter infectious = FALSE, both future and past events are considered and the estimator is:

K^*(u,v) = 1/|SxT| sum_{i=1,...,n} sum_{j=1,...,n; j \neq j} 1/wij 1/(lambda(xi)lambda(xj)) 1{uij <= u} 1{|tj - ti| <= v}

where 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.

A list containing:

`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.

Edith Gabriel <edith.gabriel@univ-avignon.fr>

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

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

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.

## 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 STIK function
u <- seq(0,10,by=1)
v <- seq(0,15,by=1)
stik <- STIKhat(xyt=FMD, s.region=northcumbria/1000,t.region=c(1,200), 
lambda=mhat*mut/dim(FMD)[1], dist=u, times=v, infectious=TRUE)

# plotting the estimation
plotK(stik)
plotK(stik,persp=T,theta=-65,phi=35)

## End(Not run)