The Z_r averaging transform in R; very useful for studying the statistical properties of sparse data

Zr.R

# Z_r (or "averaging") transform functions, based on:
# 
# Kenneth W. Church and William A. Gale. 1991. A comparison of the enhanced 
# Good-Turing and deleted estimation methods for estimating probabilities of 
# English bigrams. Computer Speech and Language 5(1):19--54
#
# Kyle Gorman <[email protected]>
# 
# Church and Gale do not say what is to be done about points at the edges. I 
# have chosen to average them with respect to only the inward facing frequency,
# which seems consistent to me with what Church and Gale had in mind. Comments
# are welcome about this choice, of course.
# 
# I am making this code available because several people have told me that it's
# not obvious.
# 
# There are versions for r/n_r vectors, and one for a single integer frequency
# vector f (if it's already probabilties, multiply it out)

Zr.nr <- function(r, nr) {
    # compute a smoothed freq distribution using Z_r statistic
    zro <- nr
    zro[1] <- zro[1] / (r[2] - r[1])
    L <- length(nr)
    i <- 2
    while (i < L) {
        zro[i] <- 2 * zro[i] / (r[i + 1] - r[i - 1])
        i <- i + 1
    }
    zro[L] <- zro[L] / (r[L] - r[L - 1])
    return(zro)
}

Zr.f <- function(f) {
    # f is a vector of integer frequencies. returns a data frame for plotting 
    q  <- rle(sort(f))
    r  <- q$values
    nr <- q$lengths
    Zr <- Zr.nr(r, nr)
    return(data.frame(r, nr, Zr))
}