# [gmx-users] g_covar -ref

Tsjerk Wassenaar tsjerkw at gmail.com
Thu Mar 18 20:42:36 CET 2010

```Hi Vijaya,

> Your answer truly eludes me, unless -ref is an useless option.

Well, that's pretty close to the conclusion. The use is limited to a
few very specific cases and it might be better to hide the option from
the view of casual users.

But apparently you didn't catch the why yet. If you imagine a pendulum
going back and forth, the principal components will be commonly
determined by taking the deviations from the mean position. Then you
get two components, one for the direction of the motion and one for
the deflection from linearity, right? But now, you calculate  the
deviations not with respect to the average position, but you take the
deviations with respect to a position that is away from the mean, say
halfway to the left. Now you can follow the same procedure,
calculating a matrix of cross products for these deviations, and
diagonalizing it. But what do these results then mean? This is a neat
example to run with pen and paper... You should even be able to solve
it analytically :)

Have fun,

Tsjerk

> In any case it isn't particularly important.
> Vijaya
>
>
>
>> Date: Thu, 18 Mar 2010 19:37:40 +0100
>> Subject: Re: [gmx-users] g_covar -ref
>> From: tsjerkw at gmail.com
>> To: gmx-users at gromacs.org
>>
>> Hi Vijaya,
>>
>> I'm sorry if I didn't quite get that first sentence of yours. Did you
>> meant to start it with "I thought that ..."? Or were you trying to
>> explain me something you thought I missed?
>>
>> PCA stands for 'principal component analysis', not 'covariance
>> analysis'. For instance, PCA can be applied to correlations, and then
>> is 'correlation analysis'. SVD is a particular flavour of PCA and here
>> yields the same results as traditional PCA because the covariance
>> matrix is symmetric, but otherwise they're not strictly the same. That
>> is to say, the SVD is obtained by extracting the eigenvectors from the
>> matrices transpose(S) x S and S x transpose(S). Which are quite
>> obviously identical if S is a symmetric matrix.
>>
>> By the way, was your question regarding the -ref option answered, or
>> did the answer elude you? If the latter is the case, maybe if now you
>> feel sufficiently confident that I know a bit about PCA, you can go
>> through the answer again.
>>
>> Cheers,
>>
>> Tsjerk
>>
>> On Thu, Mar 18, 2010 at 7:01 PM, vijaya subramanian
>> <vijaya65 at hotmail.com> wrote:
>> > PCA refers to covariance analysis (though SVD gives the same results).
>> > Principal components are obtained by projecting the trajectory onto
>> >  the eigenvectors of the covariance matrix.
>> > I just wanted to know why the option -ref was offered and if it had any
>> > significance.
>> > Thanks
>> > Vijaya
>> >
>> >> Date: Thu, 18 Mar 2010 17:51:04 +0100
>> >> Subject: Re: [gmx-users] g_covar -ref
>> >> From: tsjerkw at gmail.com
>> >> To: gmx-users at gromacs.org
>> >>
>> >> Hi Vijaya,
>> >>
>> >> Well, to start with that will be something as calculating the
>> >> 'fluctuation' as sum((xi-ri)^2)/N, with xi and ri denoting the ith
>> >> atom of the conformation x and the reference structure r and the sum
>> >> is over time/observations. In the case of no variation in xi, the
>> >> value you get will still be finite, in stead of zero, as would
>> >> probably be most meaningful.
>> >> Now for the covariances, there's a bit more to it. The covariance is
>> >> the product moment of the deviations: sum((xi-ri)(xj-rj))/N. When
>> >> there is no correlation, the deviations about the mean are random and
>> >> average out to zero. But with the deviations against a reference, that
>> >> is not the case. So the results should be regarded meaningless, unless
>> >> you have a good reason for doing so, and come with a solid
>> >> justification. Okay, there may be a purpose, but I'll leave that to
>> >> your imagination :)
>> >>
>> >> Hope it helps,
>> >>
>> >> Tsjerk
>> >>
>> >> On Thu, Mar 18, 2010 at 5:33 PM, vijaya subramanian
>> >> <vijaya65 at hotmail.com> wrote:
>> >> > Hi
>> >> > Has anyone studied the effect of using different reference
>> >> > structures,
>> >> > not the average structure, when carrying out PCA.  Does it make sense
>> >> > to
>> >> > use
>> >> > a structure besides the average to calculate the covariance matrix?
>> >> > Thanks
>> >> > Vijaya
>> >> >
>> >> >
>> >> >
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>> >>
>> >>
>> >> --
>> >> Tsjerk A. Wassenaar, Ph.D.
>> >>
>> >> post-doctoral researcher
>> >> Molecular Dynamics Group
>> >> Groningen Institute for Biomolecular Research and Biotechnology
>> >> University of Groningen
>> >> The Netherlands
>> >> --
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>>
>>
>> --
>> Tsjerk A. Wassenaar, Ph.D.
>>
>> post-doctoral researcher
>> Molecular Dynamics Group
>> Groningen Institute for Biomolecular Research and Biotechnology
>> University of Groningen
>> The Netherlands
>> --
>> gmx-users mailing list gmx-users at gromacs.org
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>
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--
Tsjerk A. Wassenaar, Ph.D.

post-doctoral researcher
Molecular Dynamics Group
Groningen Institute for Biomolecular Research and Biotechnology
University of Groningen
The Netherlands

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