AMBER Archive (2004)Subject: Re: AMBER: question about covariance matrix from ptraj
From: Elijah Gregory (ejgregory_at_gmail.com)
Date: Mon Aug 09 2004  15:00:06 CDT
> First if the number of atoms is N. Then the covariance matrix is 3N by 3N
> (each atom has three cordinates). But I am wondering what format is the
> matrix. Are the three cordinates for one atom next to each other(atom1(x,y,z),
> atom2(x,y,z), ...)? Or is the correlation along each axis grouped
> together (like atom1(x), atom2(x),...atom1(y), atom2(y)...)?
Yes, it is the former. Like atom1<x,y,z>,atom2<x,y,z>,...,atomN<x,y,z>.
> Second it is not easy to study the relationship between each atom in this
> 3N by 3N matrix. So I am wondering if there is a way that I can reduce
> the matrix to N by N (it means each element represents the correlation
> between atom i and atom j; combine the seperated information along three
> axes together)?
You could try and calculate the covariance matrix in distance space. I
think there is a "dist" option for the ptraj analyze matrix command.
Equivalently, you could define one frame as the "origin" subtract this
frame from the others, and then calculate the "distance" using
sqrt(x^2 + y^2 + z^2) = r from this reference structure for each
atom. Then the molecules' covariance matrix will be only NxN. This
might make it easier to see obvious correlations between groups of
atoms, but it is still going to be difficult to conceptualize the 3D
molecule while looking at the plot of a covariance matrix.
One of the advantages with keeping the matrix 3Nx3N is that the
eigenvectors obtained can be used to project each atom onto a subspace
spanned by the eigenvector. If you were using a distance matrix, or
any other NxN matrix, how would you define the direction the vector is
pointing in three dimensional space??? More importantly, how would you
be able to tell what directions the atoms were moving in relation to
one another?
Have you read much in the literature about using the covariance matrix
to detect correlations between atoms? If so, I'd be really interested
to read them ;]
Regards,
Elijah Gregory
Center for High Performance Computing
University of Utah
P.S. I hope you got your ptraj problem worked out ok. It sounds like you did.

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