Round to Integer Keeping the Sum Fixed

Given a real numbers \(y_i\) with the particular property that \(\sum_i y_i\) is integer, find integer numbers \(x_i\) which are close to \(y_i\) (\(\left|x_i - y_i\right| < 1 \forall i\)), 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"))

Arguments

x

a numeric vector which must sum to an integer

method

character string specifying the algorithm to be used.

Details

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, \(\min_x \left\|y - x\right\|_p\), typically for all \(p \in [1,\infty]\) simultaneously, but have not bothered to find a formal proof.

Value

a numeric vector, say r, of the same length as x, but with integer values and fulfulling sum(r) == sum(x).

Author

Martin Maechler, November 2007

References

Michel Balinski and H. Peyton Young (1982) Fair Representation: Meeting the Ideal of One Man, One Vote;

https://en.wikipedia.org/wiki/Apportionment_paradox

https://www.ams.org/samplings/feature-column/fcarc-apportionii3

See also

round etc

Examples

## 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
#> [1] 17
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|"]))))
#>            x r2i(x)  resid |res|
#>  [1,] -1.400     -1 -0.400 0.400
#>  [2,] -1.000     -1  0.000 0.000
#>  [3,]  0.244      0  0.244 0.244
#>  [4,]  0.493      1 -0.507 0.507
#>  [5,]  1.222      1  0.222 0.222
#>  [6,]  1.222      1  0.222 0.222
#>  [7,]  2.000      2  0.000 0.000
#>  [8,]  2.000      2  0.000 0.000
#>  [9,]  2.200      2  0.200 0.200
#> [10,]  2.444      2  0.444 0.444
#> [11,]  3.625      4 -0.375 0.375
#> [12,]  3.950      4 -0.050 0.050
#> [13,] 17.000     17  0.000 2.664

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)
#> sum(y) = -12 
#> [1] TRUE

## 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)
#> sum(y) = 1128 
#> [1] TRUE

if (FALSE) { # \dontrun{
for(i in 1:1000)
    stopifnot(chk(makeIntSum(rpois(100, 10) +
                             c(runif(75, min= 0, max=.2),
                               runif(25, min=.5, max=.9)))))
} # }