[gmx-users] eigenvalues and number of frames

[Daan van Aalten]

Dear Jose

The principle of this is very simple: you can't calculate the value of
more parameters than the number of observations you put in. Thus, even
though you might have a 300 dimensional space (with a theoretical system
of 100 atoms), if you only put in 25 structures it is possible to
calculate at the VERY most 25 eigenvectors. If you put in more than 300
structures the eigenvectors become more reliable as in fact your system is
then overdetermined. This has also been explained in a paper which
describes calculation of eigenvectors on very underdetermined systems
(ensembles of PDB files, no simulation) in :

D.M.F. van Aalten, D.A. Conn, B.L. de Groot, H.J.C. Berendsen, J.B.C.
Findlay and A. Amadei, "Protein dynamics derived from clusters of crystal
structures", Biophys. J. (1997), 73, 2891-2896.

warmest regards

Daan


##############################################################################

Dr. Daan van Aalten                    Wellcome Trust RCD Fellow
Wellcome Trust Biocentre, Dow Street   TEL: ++ 44 1382 344979
Div. of Biol.Chem. & Mol.Microbiology  FAX: ++ 44 1382 345764
School of Life Sciences                E-mail: dava at davapc1.bioch.dundee.ac.uk
Univ. of Dundee, Dundee DD1 5EH, UK    WWW: http://davapc1.bioch.dundee.ac.uk

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On Fri, 5 Apr 2002, Jose D Faraldo-Gomez wrote:

> > > Can anyone explain to me why when I use g_covar to build the >
> >covariance matrix and obtain the corresponding eigenvalues, the
> > > number of (clearly)non-zero eigenvalues appear to depend on the > number
> >of frames read (if this is smaller than 3N)?
>
> >You can think of the set of eigenvectors (and corresponding
> >eigenvalues) as a set of difference vectors beteween extreme
> >conformations. Obviously, there are only so many differences
> >between so many conformations, so you will only have as many
> >non-zero eigenvalues as there are conformations (or, actually
> >one less, I think).
>
> Thanks Anton/Bert; I think I can come to terms with this idea of the number
> of eigenvectors being equal to nframes-1 when nframes < 3N; though I'm not
> so sure how to apply this view when nframes > 3N, but in any case...
>
> The thing that is bothering me is that I don't see how this condition
> appears in the diagonalization process in gromacs, so I must be not
> understanding either the algebra or the code.
>
> For example, when I look at a 4ns interval using 201 and 101 frames, and I
> compare the covariance matrices (dumped by g_covar_d -debug), I don't find
> very large differences; for instance the ratio of the diagonal elements
> C(ii; 101f)/C(ii; 201 fr) is on average 0.98 +/- 0.08...
>
> So where is the trick? Are the covariance matrices really different enough
> to give lists of precisely 100 and 200 eigenvalues? This is hard to believe
> (though I'm not very good at maths)...
>
> By looking at the source code, I understand the matrices are passed on to
> the ql77 routine, which diagonalizes them and returns the eigenvalues and
> eigenvectors (and ql77 doesn't know what the number of frames is). I was
> looking for some sort of further maths being done on the eigenvalues, but I
> couldn't see anything in the code (in contrast, the eigenvectors lists are
> indeed truncated at nframes by default if nframes < ndim). Thus, is this
> condition imposed somewhere else? If so, where?
>
> Thanks a lot once again,
>
> Jose
>
>
>
>
>
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