[R-pkg-devel] mvrnorm, eigen, tests, and R CMD check
Ben Bolker
bbolker @ending from gm@il@com
Thu May 17 21:55:27 CEST 2018
There have been various comments in this thread (by me, and I think
Duncan Murdoch) about how you can identify the platform you're running
on (some combination of .Platform and/or R.Version()) and use it to
write conditional statements so that your tests will only be compared
with reference values that were generated on the same platform ... did
those get through? Did they make sense?
On Thu, May 17, 2018 at 3:30 PM, Kevin Coombes
<kevin.r.coombes at gmail.com> wrote:
> Yes; I'm pretty sure that it is exactly the repeated eigenvalues that are
> the issue. The matrices I am using are all nonsingular, and the various
> algorithms have no problem computing the eigenvalues correctly (up to
> numerical errors that I can bound and thus account for on tests by rounding
> appropriately). But an eigenvalue of multiplicity M has an M-dimensional
> eigenspace with no preferred basis. So, any M-dimensional (unitary) change
> of basis is permitted. That's what give rise to the lack of reproducibility
> across architectures. The choice of basis appears to use different
> heuristics on 32-bit windows than on 64-bit Windows or Linux machines. As a
> result, I can't include the tests I'd like as part of a CRAN submission.
>
> On Thu, May 17, 2018, 2:29 PM William Dunlap <wdunlap at tibco.com> wrote:
>
>> Your explanation needs to be a bit more general in the case of identical
>> eigenvalues - each distinct eigenvalue has an associated subspace, whose
>> dimension is the number repeats of that eigenvalue and the eigenvectors for
>> that eigenvalue are an orthonormal basis for that subspace. (With no
>> repeated eigenvalues this gives your 'unique up to sign'.)
>>
>> E.g., for the following 5x5 matrix with two eigenvalues of 1 and two of 0
>>
>> > x <- tcrossprod( cbind(c(1,0,0,0,1),c(0,1,0,0,1),c(0,0,1,0,1)) )
>> > x
>> [,1] [,2] [,3] [,4] [,5]
>> [1,] 1 0 0 0 1
>> [2,] 0 1 0 0 1
>> [3,] 0 0 1 0 1
>> [4,] 0 0 0 0 0
>> [5,] 1 1 1 0 3
>> the following give valid but different (by more than sign) eigen vectors
>>
>> e1 <- structure(list(values = c(4, 1, 0.999999999999999, 0,
>> -2.22044607159862e-16
>> ), vectors = structure(c(-0.288675134594813, -0.288675134594813,
>> -0.288675134594813, 0, -0.866025403784439, 0, 0.707106781186547,
>> -0.707106781186547, 0, 0, 0.816496580927726, -0.408248290463863,
>> -0.408248290463863, 0, -6.10622663543836e-16, 0, 0, 0, -1, 0,
>> -0.5, -0.5, -0.5, 0, 0.5), .Dim = c(5L, 5L))), .Names = c("values",
>> "vectors"), class = "eigen")
>> e2 <- structure(list(values = c(4, 1, 1, 0, -2.29037708937563e-16),
>> vectors = structure(c(0.288675134594813, 0.288675134594813,
>> 0.288675134594813, 0, 0.866025403784438, -0.784437556312061,
>> 0.588415847923579, 0.196021708388481, 0, 4.46410900710223e-17,
>> 0.22654886208902, 0.566068420404321, -0.79261728249334, 0,
>> -1.11244069540181e-16, 0, 0, 0, -1, 0, -0.5, -0.5, -0.5,
>> 0, 0.5), .Dim = c(5L, 5L))), .Names = c("values", "vectors"
>> ), class = "eigen")
>>
>> I.e.,
>> > all.equal(crossprod(e1$vectors), diag(5), tol=0)
>> [1] "Mean relative difference: 1.407255e-15"
>> > all.equal(crossprod(e2$vectors), diag(5), tol=0)
>> [1] "Mean relative difference: 3.856478e-15"
>> > all.equal(e1$vectors %*% diag(e1$values) %*% t(e1$vectors), x, tol=0)
>> [1] "Mean relative difference: 1.110223e-15"
>> > all.equal(e2$vectors %*% diag(e2$values) %*% t(e2$vectors), x, tol=0)
>> [1] "Mean relative difference: 9.069735e-16"
>>
>> > e1$vectors
>> [,1] [,2] [,3] [,4] [,5]
>> [1,] -0.2886751 0.0000000 8.164966e-01 0 -0.5
>> [2,] -0.2886751 0.7071068 -4.082483e-01 0 -0.5
>> [3,] -0.2886751 -0.7071068 -4.082483e-01 0 -0.5
>> [4,] 0.0000000 0.0000000 0.000000e+00 -1 0.0
>> [5,] -0.8660254 0.0000000 -6.106227e-16 0 0.5
>> > e2$vectors
>> [,1] [,2] [,3] [,4] [,5]
>> [1,] 0.2886751 -7.844376e-01 2.265489e-01 0 -0.5
>> [2,] 0.2886751 5.884158e-01 5.660684e-01 0 -0.5
>> [3,] 0.2886751 1.960217e-01 -7.926173e-01 0 -0.5
>> [4,] 0.0000000 0.000000e+00 0.000000e+00 -1 0.0
>> [5,] 0.8660254 4.464109e-17 -1.112441e-16 0 0.5
>>
>>
>>
>>
>>
>> Bill Dunlap
>> TIBCO Software
>> wdunlap tibco.com
>>
>> On Thu, May 17, 2018 at 10:14 AM, Martin Maechler <
>> maechler at stat.math.ethz.ch> wrote:
>>
>>> >>>>> Duncan Murdoch ....
>>> >>>>> on Thu, 17 May 2018 12:13:01 -0400 writes:
>>>
>>> > On 17/05/2018 11:53 AM, Martin Maechler wrote:
>>> >>>>>>> Kevin Coombes ... on Thu, 17
>>> >>>>>>> May 2018 11:21:23 -0400 writes:
>>>
>>> >> [..................]
>>>
>>> >> > [3] Should the documentation (man page) for "eigen" or
>>> >> > "mvrnorm" include a warning that the results can change
>>> >> > from machine to machine (or between things like 32-bit and
>>> >> > 64-bit R on the same machine) because of difference in
>>> >> > linear algebra modules? (Possibly including the statement
>>> >> > that "set.seed" won't save you.)
>>>
>>> >> The problem is that most (young?) people do not read help
>>> >> pages anymore.
>>> >>
>>> >> help(eigen) has contained the following text for years,
>>> >> and in spite of your good analysis of the problem you
>>> >> seem to not have noticed the last semi-paragraph:
>>> >>
>>> >>> Value:
>>> >>>
>>> >>> The spectral decomposition of ‘x’ is returned as a list
>>> >>> with components
>>> >>>
>>> >>> values: a vector containing the p eigenvalues of ‘x’,
>>> >>> sorted in _decreasing_ order, according to ‘Mod(values)’
>>> >>> in the asymmetric case when they might be complex (even
>>> >>> for real matrices). For real asymmetric matrices the
>>> >>> vector will be complex only if complex conjugate pairs
>>> >>> of eigenvalues are detected.
>>> >>>
>>> >>> vectors: either a p * p matrix whose columns contain the
>>> >>> eigenvectors of ‘x’, or ‘NULL’ if ‘only.values’ is
>>> >>> ‘TRUE’. The vectors are normalized to unit length.
>>> >>>
>>> >>> Recall that the eigenvectors are only defined up to a
>>> >>> constant: even when the length is specified they are
>>> >>> still only defined up to a scalar of modulus one (the
>>> >>> sign for real matrices).
>>> >>
>>> >> It's not a warning but a "recall that" .. maybe because
>>> >> the author already assumed that only thorough users would
>>> >> read that and for them it would be a recall of something
>>> >> they'd have learned *and* not entirely forgotten since
>>> >> ;-)
>>> >>
>>>
>>> > I don't think you're really being fair here: the text in
>>> > ?eigen doesn't make clear that eigenvector values are not
>>> > reproducible even within the same version of R, and
>>> > there's nothing in ?mvrnorm to suggest it doesn't give
>>> > reproducible results.
>>>
>>> Ok, I'm sorry ... I definitely did not want to be unfair.
>>>
>>> I've always thought the remark in eigen was sufficient, but I'm
>>> probably wrong and we should add text explaining that it
>>> practically means that eigenvectors are only defined up to sign
>>> switches (in the real case) and hence results depend on the
>>> underlying {Lapack + BLAS} libraries and therefore are platform
>>> dependent.
>>>
>>> Even further, we could consider (optionally, by default FALSE)
>>> using defining a deterministic scheme for postprocessing the current
>>> output of eigen such that at least for the good cases where all
>>> eigenspaces are 1-dimensional, the postprocessing would result
>>> in reproducible signs, by e.g., ensuring the first non-zero
>>> entry of each eigenvector to be positive.
>>>
>>> MASS::mvrnorm() and mvtnorm::rmvnorm() both use "eigen",
>>> whereas mvtnorm::rmvnorm() *does* have method = "chol" which
>>> AFAIK does not suffer from such problems.
>>>
>>> OTOH, the help page of MASS::mvrnorm() mentions the Cholesky
>>> alternative but prefers eigen for better stability (without
>>> saying more).
>>>
>>> In spite of that, my personal recommendation would be to use
>>>
>>> mvtnorm::rmvnorm(.., method = "chol")
>>>
>>> { or the 2-3 lines of R code to the same thing without an extra package,
>>> just using rnorm(), chol() and simple matrix operations }
>>>
>>> because in simulations I'd expect the var-cov matrix Sigma to
>>> be far enough away from singular for chol() to be stable.
>>>
>>> Martin
>>>
>>> ______________________________________________
>>> R-package-devel at r-project.org mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-package-devel
>>>
>>
>>
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