## A function to find the nearest neighbor of a point
require (class) 
nearest <- function (X, n, k=3) 
## Find k nearest neighbors of X[n, ] in the data frame 
## or matrix X, utilizing function knn1 k-times. 
{ 
    N <- nrow(X) 
    # inds contains the indices of nearest neighbors 
    inds <- c(n); i <- 0 
    while (i < k) { 
        # use knn1 for one index... 
        j <- as.integer(knn1(X [-inds, ], X[n, ], 1:(N-length(inds)))) 
        # ...and change to true index of neighbor 
        inds <- c(inds, setdiff(1:N, inds)[j]) 
        i <- i+1 
    } 
    # return nearest neighbor indices (without n, of course) 
    return(inds[-1]) 
} 


## A function to calculate the spatial statistic discussed in the midterm
spat.stat <- function(X){
n <- nrow(X)

Out <- real()
for(i in 1:n){
Out[i] <- sqrt( (X[i,1]- X[nearest(X, i, 1),1])^2 + (X[i,2]- X[nearest(X,i,1),2])^2)
}
Out <- mean(Out)
return(Out)
}