Testing for CSR

Distance-based measure using events locations: however there are \(n(n-1)/2\) inter-event distances to check for the \(n\) nearest neighbour distances so is computationally intensive. To deal with computationally this a tesselation or triangulation is used usually.

Delauney/Voronoi tesslation: the Dirichlet tile associated with point \(Z(s_i)\) is the region of space that is closer to \(Z(s_i)\) than to any other point in \(Z(s)\).

Delauney triangulation: 1) Construct the Dirchlet/Voronoi tesselation, 2) Two points are Delauney neighbours if their Dirchlet/Voronoi tiles share a common boundary. This results in a tesselation of disjoint triangles (the union of the triangles is the convex hull).

Testing for CSR

plot(murchison$gold)

nm <- nndist.ppp(murchison$gold)
#nm
summary(nm)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   899.3  1810.4  2174.8  3515.4  2892.0 36118.0

Testing for CSR

plot(austates)

a <- austates

Testing for CSR

ld <- as.tess(austates)
ld

## Tessellation
## Tiles are irregular polygons
## 7 tiles (irregular windows)
## window: polygonal boundary
## enclosing rectangle: [113.19392, 153.6692] x [-43.59316, -10.93156] units

Testing for CSR

plot(ld)

Testing for CSR

summary(austates)

##        Length Class  Mode     
## type   1      -none- character
## window 5      owin   list     
## tiles  7      -none- list     
## n      1      -none- numeric

Testing for CSR

x <- load.image("scale1to10.jpg") 
#Reads in the image as a matrix (uses jpeg package)
t <- threshold(x, 0.5) 
#returns a binary image (matrix) (uses EBImage package) 
plot(t)

Testing for CSR

m <- matrix(t, ncol = ncol(x), nrow = nrow(x))  #blank matrix
ss <- ncol(x)
m <- m[,ncol(m):1]
z <- NULL
for (i in 1:ncol(t)) {
   for (j in 1:nrow(t)) {
        if (m[i,j] == 1){
            z <- rbind(z,c(i,j))
        } } }
ww <- owin(c(0,ncol(t)), c(0,nrow(t)))
d <- as.ppp(z,ww)
dsub <-d[owin(c(0,100), c(0,100))]

Testing for CSR

plot(dsub, pch = 20, cex = 0.1 )

Testing for CSR

dt <- tess(dsub, image = m[1:100,1:100], 
           window = owin(c(0,100), c(0,100) ))
plot(dt)

Testing for CSR

tiles(dt)

## List of spatial objects
## 
## FALSE:
## window: binary image mask
## 100 x 100 pixel array (ny, nx)
## enclosing rectangle: [0.5, 100.5] x [0.5, 100.5] units
## 
## TRUE:
## window: binary image mask
## 100 x 100 pixel array (ny, nx)
## enclosing rectangle: [0.5, 100.5] x [0.5, 100.5] units

dtd <- dirichlet(dsub)
dtd

## Tessellation
## Tiles are windows of general type
## 1894 tiles (irregular windows)
## window: rectangle = [0, 100] x [0, 100] units

Testing for CSR

plot(dtd)

\(F\)-function: the empty-space function

\(F(r) = P[\rho(a,x) \le r]\): the probability of observing at least one point closer than \(r\) to the arbitrary point \(a\) (not necessarily a point of the pattern). If the estimate is smaller than CSR (\(F(r) = 1- e^{-\pi \lambda r^2}\)) then the points are clustered, if larger than CSR then the points are regular.

\(1-F (r)\) is the probability that a randomly placed disk with radius \(r\) does not contain a point.

\(F\)-function: the empty-space function

ff <- Fest(dsub, correction = "none")
f <- Fest(dsub)
plot(f)

plot(ff)

\(F\)-function: the empty-space function

ffci <- varblock(dsub, fun = Fest, correction = "none")
plot(ffci)

#ffciloh <- lohboot(dsub, fun = Fest, correction = "none")

\(F\)-function: the empty-space function

ff <- Fest(dsub)
plot(ff)

\(F\)-function: the empty-space function

plot(density.ppp(dsub))

ffinhom <- Finhom(dsub)
plot(ffinhom)

\(F\)-function: the empty-space function

ffinhomci <- varblock(dsub, fun = Finhom, correction = "none")
plot(ffinhomci)

\(G\)-function: nearest-neighbour distance function

An empirical estimate of the probability that the nearest-neighbour distance is at most \(r_0\): \[\hat{G}(r_0) = \frac{\#(r_i-r_0)}{n}.\] If the estimate is larger than CSR (\(G(r) = 1- e^{-\pi \lambda r^2}\)) then the points are clustered, if smaller than CSR then the points are regular.

\(1-G(r)\) is the probability that a disk with radius \(r\) centered at a randomly selected point does not contain a further point.

\(G\)-function: nearest-neighbour distance function

gg <- Gest(dsub, correction = "none")
plot(gg)

\(G\)-function: nearest-neighbour distance function

ggci <- varblock(dsub, fun = Gest, correction = "none")
plot(ggci)

\(G\)-function: nearest-neighbour distance function

gginhom <- Ginhom(dsub)
plot(gginhom)

\(G\)-function: nearest-neighbour distance function

gginhomci <- varblock(dsub, fun = Ginhom, correction = "none")
plot(gginhomci)