[gmx-users] definition of "eigenvector" in gromacs

[Antonio Baptista]

Tsjerk, that's one of the coolest explanations of PCA I've ever read! :)

Just one additional comment for Brett: Tsjerk's example assumes implicitly 
that your hands are actually moving parallel to the desk on average. On my 
desk, the meal is usually on top of the pile of papers that mysteriously 
keeps growing to the left of my keyboard. So, my hands don't move parallel 
to the desk, but along an inclined plane that raises from the keyboard to 
the meal position and is slightly tilted towards me (yeah, I know I should 
sit straight...). Therefore, the average position of my hands and their 
direction of major extent (eigenvector 1) should be marked on that plane, 
not on the desk, and the second direction (eigenvector 2) is a line also 
on that plane drawn perpendicularly to the first. As in Tsjerk's example, 
if I project every position of my hands onto this resulting PCA plane I 
will probably capture most of their motion, without having to consider the 
third direction perpendicular to the plane.

My point is that the subspace obtained from PCA is often different from 
the one that seems more "natural" to us (which was perhaps not obvious in 
Tsjerk's example). Although my desk is the "natural" reference frame when 
I'm working, my hands' motion can be more easily described in terms of 
positions on the plane I've just described, since I roughly need just 2 
coordinates instead of the original 3. That's the aim of PCA -- to 
describe the original motion using fewer coordinates.

Incidentally, this also shows that a messy desk is scientifically more 
fruitful than a tidy one... ;)

Cheers,
Antonio

On Mon, 9 Mar 2015, Tsjerk Wassenaar wrote:

> Hi Brett,
>
> Let's say you're sitting at your _desk_ writing that paper with a deadline
> yesterday and you put a quick _meal_ next to you, wondering why on _earth_
> you keep up with this. Your hands are moving between the meal and the
> keyboard. You notice that the average position of your hands is somewhere
> between the two and mark the mean position on your desk. Then you draw a
> line through it that corresponds to the major extent of the motion of your
> hands, and write 'eigenvector 1' along it. You add a line through the
> average position, perfectly perpendicular to the first, and write
> 'eigenvector 2' along it. Now you can project every position of your hands
> onto your desk, giving it an 'eigenvector 1' coordinate (or score) and an
> 'eigenvector 2' coordinate (or score). You notice that it's only part of
> the total motion, as you neglect the height, which will be a third line,
> perpendicular to the desk.
>
> You can look at it a bit differently and say that your desk is the subspace
> of your real space, spanned by the two perpendicular vectors, which
> together describe most of your hand's motion.
>
> I hope this makes some sense :)
>
> Cheers,
>
> Tsjerk
>
>
>
>
>
> On Mon, Mar 9, 2015 at 2:33 AM, Brett <brettliu123 at 163.com> wrote:
>
>> Dear All,
>>
>> Recently I have read a gromacs related article which contain a sentence
>> "Trajectories pojected on the first two eigenvectors". Will you please tell
>> me the meaning of "eigenvector" and "Trajectories pojected on the first two
>> eigenvectors"? Will you please also explain it to me using common-used
>> words simple as desk, meal, earth, etc?
>>
>> Brett
>>
>>
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>
>
> -- 
> Tsjerk A. Wassenaar, Ph.D.
> -- 
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--
Antonio M. Baptista
Instituto de Tecnologia Quimica e Biologica, Universidade Nova de Lisboa
Av. da Republica - EAN, 2780-157 Oeiras, Portugal
phone: +351-214469619         email: baptista at itqb.unl.pt
fax:   +351-214411277         WWW:   http://www.itqb.unl.pt/~baptista
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