Find the area of a region using monte carlo methods - spatial

Question

I am interested in finding the area of the dark gray region, in r I have used the area function of spatstat package but is very time consuming since I need to run a mh algorithm, I have read that I can calculate this area using monte carlo approach but I do not know how to proceed, please someone can help me?
some code example from spatstat book for the plot and for the area:
source("figurelayout.R")
source("startup.R")
llibrary(spatstat)
requireversion(spatstat, "1.41-1.073")
W <- as.owin(swedishpines)
x <- c(28,29,55,60,66)
y <- c(70,38,32,72,59)
X <- ppp(x=x,y=y, window = W)
u <- list(x=48,y=50)
u <- as.ppp(u, W)
rad <- 14
Xplusr <- dilation(X, rad) #
uplusr <- disc(rad, u) #bola de radio rad con centro en u
ovlap <- intersect.owin(uplusr, Xplusr)
B<-setminus.owin( disc( rad, c(u$x, u$y) ), Xplusr)
AIdemo <- layered(W,
ovlap,
Xplusr,
uplusr,
X,
u,B)
layerplotargs(AIdemo) <- list(list(),
list(col="darkgrey", border=NA),
list(lwd=2),
list(lwd=2, lty=2),
list(pch=16),
list(pch=3),
list(col="red", border=NA))
###################################################
### code chunk number 98: Unit.Rnw:3-5
###################################################
newplot(6, 0.7)
setmargins(0)
###################################################
### code chunk number 99: 13gibbs.Rnw:3214-3215
###################################################
plot(AIdemo, main="")
plot(setminus.owin( disc( rad, c(u$x, u$y) ), Xplusr))
area<-area(B)

Answer 1

What you are asking is a simple Monte Carlo application. You will need to draw uniform samples across the whole region, and check the proportion of points selected within your region.
To check if a point lies within the region, you can describe your region with the center-radius form of the circle equation (your region is just a circle).
Follow the link:
http://geekyisawesome.blogspot.com.au/2013/03/montecarlo-method-of-finding-area-of.html

Answer 2

Your question title suggests that you want to use Monte Carlo, but the question itself just says "I am interested in finding the area of the dark gray region ...". Another way to calculate this area in spatstat is to use the function areaLoss.
Continuing your example (which I guess is really copyrighted example code from the companion website of the spatstat book):
Y <- superimpose(u, X) # Centres of all circles
areaLoss(Y, rad, exact = TRUE) # Remaining area in each circle
areaLoss(Y, rad, exact = TRUE, subset = 1) # Only circle of interest
Depending on how many overlapping circles you have this may be faster than what you currently are doing. Also depending on the number of circles setting exact = FALSE may be faster.
If you really wanted to estimate the area of B by simple Monte Carlo it can be done in a few lines of code:
Bbox <- as.rectangle(B)
points <- runifpoint(1000, Bbox)
insideB <- inside.owin(points, w = B)
fraction <- mean(insideB)
area(Bbox) * fraction

Answer 3

Thanks you for your comments, here is the code for the solution :)
x.min <- min(W$xrange[1])
x.max <- max(W$xrange[2])
y.min <- min(W$yrange[1])
y.max <- max(W$yrange[2])
n.sims <- 1000
random.points <- data.frame(Q=runif(n=n.sims,x.min,x.max),
DOC=runif(n=n.sims,y.min,y.max))
random.points<-data.frame(Xinterval=runif(n=n.sims,x.min,x.max),Yinterval=runif(n=n.sims,y.min,y.max))
contador<-0
for ( ii in 1:n.sims ) {
if(inside.owin(random.points$Xinterval[ii], random.points$Yinterval[ii], w=B)
){
contador<-contador+1
}
}
estimatedArea<-area(W)*contador/n.sims