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## The Lorenz curve is a continuous piecewise linear function
## representing the distribution of income or wealth.
## p_i=i/n, i=1,...,n
## L_i=sum_{j=1}^{i} x_j / sum_{j=1}^{n} x_j
## p_0=L_0=0
lorenz <-
function(x, n = rep(1, length(x)), na.last=TRUE)
{
if (any(x < 0))
stop("x must not be < 0")
o <- order(x, na.last=na.last)
xo <- xo0 <- x[o]
no <- n[o]
xo <- xo * no
# p <- seq_len(length(xo)) / length(xo)
p <- cumsum(no) / sum(no)
L <- cumsum(xo) / sum(xo)
p <- c(0, p)
L <- c(0, L)
J <- p - L
G <- sum(xo * seq_len(length(xo)))
G <- 2 * G / (length(xo) * sum(xo))
G <- G - 1 - (1 / length(xo))
m1 <- which.max(J)
out <- cbind(p=p, L=L, x=c(0, xo0))
## x: habitat suitability cutoff is the back scaled
## L value (original x) from the graph
## L: cumulative distribution of the metric of interest
## p: cumulative distribution of the available population
## S: asymmetry is the sum of x and y coordinates
## at the point of slope 1 (symmetry: S=1)
## G: Gini coefficient of 0 mean perfect equality,
## values close to 1 indicate high inequality.
## Youden index
attr(out, "summary") <- c(
"t" = unname(m1),
"x[t]" = unname(xo0[m1 - 1L]),
"p[t]" = unname(p[m1]),
"L[t]" = unname(L[m1]),
"G" = G,
"S" = unname(L[m1] + p[m1]),
"J" = max(J))
class(out) <- c("lorenz", "matrix")
out
}
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