iLISTA: i-th product density LISTA function
In frajaroco/pdLISTA: LISTA functions

Description Usage Arguments Details Value Author(s) References Examples

Computes i-th edge-corrected kernel estimator of the product density LISTA function.

1 2	iLISTA(xyt, i, s.region, t.region, ds, dt, ks = "epanech", hs, kt = "epanech", ht, correction = TRUE)

`xyt`	Spatial-temporal coordinates (x,y,t) of the point pattern.
`i`	Point position for which you want to calculate a LISTA function.
`s.region`	A two-column matrix specifying a polygonal region containing all data locations. If `s.region` is missing, the Ripley-Rasson estimate convex spatial domain 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.
`ds`	A vector of distances `u` at which ρ^{(2)i}(u,v) is computed.
`dt`	A vector of distances `v` at which ρ^{(2)i}(u,v) is computed.
`ks`	A kernel function for the spatial distances. The default is `"epanech"` the Epanechnikov kernel. It can also be `"box"` kernel, or `"biweight"`.
`hs`	A bandwidth of the kernel function `ks`.
`kt`	A kernel function for the temporal distances. The default is `"epanech"` the Epanechnikov kernel. It can also be `"box"` kernel, or `"biweight"`.
`ht`	A bandwidth of the kernel function `kt`.
`correction`	It is `TRUE` by default and the Ripley's isotropic edge-correction weights are computed. If it is `FALSE` the estimated is without edge-correction.

An individual product density LISTA functions ρ^{(2)i}(.,.) is a puntual contribution coming from (u_i,t_i) to the global estimator of the second-order product density ρ^{(2)}(.,.), and may provide local description of structure in the spatio-temporal point data (e.g., determining events with similar local structure through dissimilarity measures of the individual LISTA functions), for more details see Cressie and Collins (2001).

A list containing:

hlista: A matrix containing the values of the estimation of \widehat{ρ}^{(2)i}(r,t) for the point i of the process by rows.
ds: Vector of distances u at which ρ^{(2)i}(r,t) is computed under the restriction 0<ε<r.
dt: Vector of distances v at which ρ^{(2)i}(r,t) is computed under the restriction 0<δ<t.
kernel: A vector of names and bandwidth of the spatial and temporal kernel.

Francisco J. Rodriguez-Cortes <cortesf@uji.es> https://fjrodriguezcortes.wordpress.com

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.

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

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

# Realisations of the homogeneous spatio-temporal Poisson processes
stp <- rpp(100)$xyt
# Generated spatio-temporal point pattern
plot(stp)

# Randomly selected point and its product density LISTA 
fixed <-sample(1:length(stp[,1]),1)
 
# Estimation of the individual i-th product density LISTA function 
out <- iLISTA(stp,i=fixed)
z1 <- out$hlista

# Spatio-temporal LISTA surface
par(mfrow=c(1,1))
persp(out$ds,out$dt,z1,theta=-45,phi=30,zlim=range(z1,na.rm=TRUE),expand=0.7,ticktype="detailed",xlab="r = distance",ylab="t = time",zlab="",cex.axis=0.7, cex.lab=0.7)
contour(out$ds,out$dt,z1,drawlabels=TRUE,axes=TRUE,xlab="r = distance",ylab="t = time",cex.axis=0.7, cex.lab=0.7)

## End(Not run)
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