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I'd like to fit integer coefficients, e.g. summing to 10, to a regression equation. The absolute values of the coefficients (i.e. predicted y) aren't important, I just want to retain the appropriate relative values. The use case is for an easily interpretable scoring system.

For example, this regression yields the following coefficients (ignoring the intercept):

set.seed(0)
y <- rnorm(100)
x <- matrix(rnorm(300), ncol=3)
m <- lm(y ~ x)
(coef <- m$coefficients[-1])
#          x1          x2          x3 
#  0.12100965  0.05506511  0.14708549 

Rounding with the below code yields a rounding error (sums to 11):

round(10 * coef / sum(coef))
# x1 x2 x3 
#  4  2  5 

A method like this also doesn't guarantee maximally similar weights to the regression equation.

This was asked here without satisfactory answers, and might be addressed in this paywalled research paper.

Edit: looks like https://stackoverflow.com/questions/792460/how-to-round-floats-to-integers-while-preserving-their-sum may be able to help minimize the roundoff error. If my question is further specified as minimizing the error of a predicted (scaled) y, I'm unsure whether this is an equivalent optimization.

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  • 2
    $\begingroup$ Have you considered integer linear programming? $\endgroup$ – Matthew Drury Sep 22 '15 at 21:14
  • $\begingroup$ I haven't, that could be a good approach. I'll take a stab if I get a chance. $\endgroup$ – Max Ghenis Sep 22 '15 at 21:22
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We can apply the answer to Round vector of numerics to integer while preserving their sum:

smart.round <- function(x) {
  y <- floor(x)
  indices <- tail(order(x-y), round(sum(x)) - sum(y))
  y[indices] <- y[indices] + 1
  y
}

At that point similar logic gets a reasonable answer:

# Setup from question
set.seed(0)
y <- rnorm(100)
x <- matrix(rnorm(300), ncol=3)
m <- lm(y ~ x)
coef <- m$coefficients[-1]
# Answer
smart.round(10 * coef / sum(coef))
# x1 x2 x3 
#  4  2  4

I don't know whether this also minimizes the error of a predicted y, but it does yield something feasible.

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