[R] Large determinant problem

[Prof Brian Ripley]

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
>
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> 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