[Rd] Computing means, variances and sums

From: Prof Brian Ripley <ripley_at_stats.ox.ac.uk>
Date: Sun 19 Feb 2006 - 10:58:56 GMT

There has been a recent thread on R-help on this, which resurrected concepts from bug reports PR#1228 and PR#6743. Since the discussion has included a lot of erroneous 'information' based on misunderstandings of floating-point computations, this is an attempt to set the record straight and explain the solutions adopted.

The problem was that var(rep(0.02, 10)) was observed to be (on some machines) about 1e-35. This can easily be explained.

0.02 is not an exactly repesented binary fraction. Repeatedly adding the represented quantity makes increasing rounding errors relative to the exact computation, so (on Sparc Solaris)

> var(rep(0.02, 10))

[1] 1.337451e-35
> options(digits=18)
> sum(rep(0.02, 10))

[1] 0.19999999999999998
> sum(rep(0.02, 10)) -0.2

[1] -2.775557561562891e-17
> sum(rep(0.02, 10))/10 -0.02

[1] -3.469446951953614e-18
> z <- sum(rep(0.02, 10))/10 -0.02
> 10*z^2/9
[1] 1.3374513502689138e-35

and so the non-zero variance is arising from (x[i] - mean) being non-zero. (I did check that was what was happening at C level.)

There has been talk of other ways to arrange these computations, for example Kahan summation and Welford's algorithm (see Chan & Lewis, 1979, CACM 22, 526-531 and references therein). However, R already used the two-pass algorithm which is the most accurate (in terms of error bounds) in that reference.

Why are most people seeing 0? Because the way computation is done in modern FPUs is not the computation analysed in early numerical analysis papers, including in Chan & Lewis. First, all FPUs that I am aware of allow the use of guard digits, effectively doing intermediate computations to one more bit than required. And many use extended precision registers for computations which they can keep in FP registers, thereby keeping another 10 or more bits. (This includes R on most OSes on ix86 CPUs, the exception being on Windows where the FPU has been reset by some other software. Typically it is not the case for RISC CPUs, e.g. Sparc.)

The use of extended precision registers invalidates the textbook comparisons of accuracy in at least two ways. First, the error analysis is different if all intermediate results are stored in extended precision. Second, the simpler the algorithm, the more intermediate results which will remain in extended precision. This means that (for example) Kahan summation is usually less accurate than direct summation on real-world FPUs.

There are at least two steps which can be done to improve accuracy. One is to ensure that intermediate results are stored in extended precision. Every R platform of which I am aware has a 'long double' type which can be used. On ix86 this is the extended precision type used internally in the FPU and so the cost is zero or close to zero, whereas on a Sparc the extra precision is more but there is some cost. The second step is to use iterative refinement, so the final part of mean.default currently is

     ## sum(int) can overflow, so convert here.
     if(is.integer(x)) x <- as.numeric(x)
     ## use one round of iterative refinement
     res <- sum(x)/n
     if(is.finite(res)) res + sum(x-res)/n else res

This is a well-known technique in numerical linear algebra, and improves the accuracy whilst doubling the cost. (This is about to be replaced by an internal function to allow the intermediate result to be stored in a long double.)

Note the is.finite(res) there. R works with the extended IEC60059 (aka IEEE754) quantities of Inf, -Inf and NaN (NA being a special NaN). The rearranged computations do not work correctly for those quantities. So although they can be more 'efficient' (in terms of flops), they have to be supplemented by some other calculation to ensure that the specials are handled correctly. Remember once again that we get both speed and accuracy advantages by keeping computations in FPU registers, so complicating the code has considerable cost.

R-devel will shortly use long doubles for critical intermediate results and iterative refinement for calculations of means. This may be slower but it would be an extreme case in which the speed difference was detectable. Higher accuracy has a cost too: there are several packages (dprep and mclust are two) whose results are critically dependent on fine details of computations and will for example infinite-loop if an optimized BLAS is used on AMD64.

The choice of algorithms in R is an eclectic mixture of accuracy and speed. When (some of) R-core decided to make use of a BLAS for e.g. matrix products this produced a large speed increase for those with optimized BLAS (and a small speed decrease for others), but it did result in lower accuracy and problems with NAs etc (and the alternative algorithms have since been added back to cover such cases). But it seems that nowadays few R users understand the notion of rounding error, and it is easier to make the computations maximally accurate than to keep explaining basic numerical analysis on R-help. (The problem is thereby just pushed farther down the line, because for example linear regression is not maximally accurate.)

Brian D. Ripley,                  ripley@stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

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Received on Sun Feb 19 22:01:39 2006

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