Round to Integer Keeping the Sum Fixed
Given a real numbers y_i with the particular property that
∑_i y_i is integer, find integer numbers x_i
which are close to y_i (
|x[i] - y[i]| < 1 for all i), and have identical “marginal”
sum, sum(x) == sum(y).
As I found later, the problem is known as “Apportionment Problem” and it is quite an old problem with several solution methods proposed historically, but only in 1982, Balinski and Young proved that there is no method that fulfills three natural desiderata.
Note that the (first) three methods currently available here were all (re?)-invented by M.Maechler, without any knowledge of the litterature. At the time of writing, I have not even checked to which (if any) of the historical methods they match.
roundfixS(x, method = c("offset-round", "round+fix", "1greedy"))x |
a numeric vector which must sum to an integer |
method |
character string specifying the algorithm to be used. |
Without hindsight, it may be surprising that all three methods give
identical results (in all situations and simulations considered),
notably that the idea of ‘mass shifting’ employed in the
iterative "1greedy" algorithm seems equivalent to the much simpler
idea used in "offset-round".
I am pretty sure that these algorithms solve the L_p optimization problem, min_x ||y - x||_p, typically for all p in [1,Inf] simultaneously, but have not bothered to find a formal proof.
a numeric vector, say r, of the same length as x, but
with integer values and fulfulling sum(r) == sum(x).
Martin Maechler, November 2007
Michel Balinski and H. Peyton Young (1982) Fair Representation: Meeting the Ideal of One Man, One Vote;
round etc
## trivial example
kk <- c(0,1,7)
stopifnot(identical(kk, roundfixS(kk))) # failed at some point
x <- c(-1.4, -1, 0.244, 0.493, 1.222, 1.222, 2, 2, 2.2, 2.444, 3.625, 3.95)
sum(x) # an integer
r <- roundfixS(x)
stopifnot(all.equal(sum(r), sum(x)))
m <- cbind(x=x, `r2i(x)` = r, resid = x - r, `|res|` = abs(x-r))
rbind(m, c(colSums(m[,1:2]), 0, sum(abs(m[,"|res|"]))))
chk <- function(y) {
cat("sum(y) =", format(S <- sum(y)),"\n")
r2 <- roundfixS(y, method="offset")
r2. <- roundfixS(y, method="round")
r2_ <- roundfixS(y, method="1g")
stopifnot(all.equal(sum(r2 ), S),
all.equal(sum(r2.), S),
all.equal(sum(r2_), S))
all(r2 == r2. & r2. == r2_) # TRUE if all give the same result
}
makeIntSum <- function(y) {
n <- length(y)
y[n] <- ceiling(y[n]) - (sum(y[-n]) %% 1)
y
}
set.seed(11)
y <- makeIntSum(rnorm(100))
chk(y)
## nastier example:
set.seed(7)
y <- makeIntSum(rpois(100, 10) + c(runif(75, min= 0, max=.2),
runif(25, min=.5, max=.9)))
chk(y)
## Not run:
for(i in 1:1000)
stopifnot(chk(makeIntSum(rpois(100, 10) +
c(runif(75, min= 0, max=.2),
runif(25, min=.5, max=.9)))))
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.