### curse of dimensionality

## Illustrate by simulation the following fact
## -----------------------------------------------------------------------
## In high dimension d :  Most points in Unit Cube are outside unit ball :
## -----------------------------------------------------------------------

n <- 100000
res <- numeric(20)
set.seed(22)
for(d in 1:20) { ## dimension 1, 2, ... , 20
    ## random points in unit cube  [-1, 1] ^ d:
    x <- matrix(runif(n*d, -1, 1), nrow = n, ncol = d)
    ## ratio of points falling inside the unit ball {x ; ||x|| <= 1 } :
    res[d] <- mean(rowSums(x^2) <= 1)
}
cbind(res)

library("sfsmisc") ## -> 'mult.fig()', ps.do()

ps.do(file = "curse.eps",height = 8)

mult.fig(mfrow = c(1,2),
         main ="relative frequency of U[0,1] obs. in unit ball")
plot(1:20,res,type = "b", xlab = "dimension d", ylab = "relative frequency")
abline(h = 0.2, col="light blue")
## zoom in :
plot(1:20,res,type = "b", ylim = c(0,0.2),
     xlab = "dimension d", ylab = "relative frequency")

ps.end()