[R] Large determinant problem

[maj at stats.waikato.ac.nz]

I tried crossprod(S) but the results were identical. The term
-0.5*log(det(S)) is  a complexity penalty meant to make it unattractive to
include too many components in a finite mixture model. This case was for a
9-component mixture. At least up to 6 components the determinant behaved
as expected and increased with the number of components.

Thanks for your comments.

> Hmm, S'S is numerically singular.  crossprod(S) would be a better way to
> compute it than crossprod(S,S) (it does use a different algorithm), but
> look at the singular values of S, which I suspect will show that S is
> numerically singular.
>
> Looks like the answer is 0.
>
>
> On Sun, 9 Dec 2007, maj at stats.waikato.ac.nz wrote:
>
>> I thought I would have another try at explaining my problem. I think
>> that
>> last time I may have buried it in irrelevant detail.
>>
>> This output should explain my dilemma:
>>
>>> dim(S)
>> [1] 1455  269
>>> summary(as.vector(S))
>>      Min.    1st Qu.     Median       Mean    3rd Qu.       Max.
>> -1.160e+04  0.000e+00  0.000e+00 -4.132e-08  0.000e+00  8.636e+03
>>> sum(as.vector(S)==0)/(1455*269)
>> [1] 0.8451794
>> # S is a large moderately sparse matrix with some large elements
>>> SS <- crossprod(S,S)
>>> (eigen(SS,only.values = TRUE)$values)[250:269]
>> [1]  9.264883e+04  5.819672e+04  5.695073e+04  1.948626e+04
>> 1.500891e+04
>> [6]  1.177034e+04  9.696327e+03  8.037049e+03  7.134058e+03
>> 1.316449e-07
>> [11]  9.077244e-08  6.417276e-08  5.046411e-08  1.998775e-08
>> -1.268081e-09
>> [16] -3.140881e-08 -4.478184e-08 -5.370730e-08 -8.507492e-08
>> -9.496699e-08
>> # S'S fails to be non-negative definite.
>>
>> I can't show you how to produce S easily but below I attempt at a
>> reproducible version of the problem:
>>
>>> set.seed(091207)
>>> X <- runif(1455*269,-1e4,1e4)
>>> p <- rbinom(1455*269,1,0.845)
>>> Y <- p*X
>>> dim(Y) <- c(1455,269)
>>> YY <- crossprod(Y,Y)
>>> (eigen(YY,only.values = TRUE)$values)[250:269]
>> [1] 17951634238 17928076223 17725528630 17647734206 17218470634
>> 16947982383
>> [7] 16728385887 16569501198 16498812174 16211312750 16127786747
>> 16006841514
>> [13] 15641955527 15472400630 15433931889 15083894866 14794357643
>> 14586969350
>> [19] 14297854542 13986819627
>> # No sign of negative eigenvalues; phenomenon must be due
>> # to special structure of S.
>> # S is a matrix of empirical parameter scores at an approximate
>> # mle for a model with 269 paramters fitted to 1455 observations.
>> # Thus, for example, its column sums are approximately zero:
>>> summary(apply(S,2,sum))
>>      Min.    1st Qu.     Median       Mean    3rd Qu.       Max.
>> -1.148e-03 -2.227e-04 -7.496e-06 -6.011e-05  7.967e-05  8.254e-04
>>
>> I'm starting to think that it may not be a good idea to attempt to
>> compute
>> large information matrices and their determinants!
>>
>> Murray Jorgensen
>>
>> ______________________________________________
>> R-help at r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide
>> http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
>>
>
> --
> Brian D. Ripley,                  ripley at 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
>
>