Re: distance between planes
- From: ellieandrogerxyzzy@xxxxxxxxxxxxxxxxxxxxxx (Roger Stafford)
- Date: Mon, 18 Jun 2007 21:08:27 GMT
In article <ef5acbb.1@xxxxxxxxxxxxxxxxxxxxxxx>, Paulina
<paulia7@xxxxxxxxxxx> wrote:
I am performing Principal Component Analysis on 16 5-d points-----------------
(projecting them onto a 3-d).
The first matrix (plane),which I gave above is defined by 3 selected
eigenvectors , which I Calculated from covariance matrix of my 16 5-d
points.
Another matrix is a different set of points, also with performed PCA
from 5-d to 3-d.
I have many sets of points like that.
My goal is to show how these sets of points are similar.
I was planning to calculate the distance between the planes. if the
distance is smaller, my point sets would be more similar.
All I need is a MATLAB syntax in how to calculate the distance
between two 3-d planes.
You haven't stated very clearly what you mean by the term "plane" here.
Apparently, for each point set you have selected the three principal
eigenvectors, and your "plane" consists of all possible linear
combinations of these eigenvectors added to the points' mean value
vector. This would be a three dimensional linear space imbedded in
five-dimensional space.
You are then asking how to find the "distance" between two such
three-dimensional spaces for two different point sets, but you don't
define what you mean by distance here. In general, two three-dimensional
linear spaces within a five-dimensional space would have a common
one-dimensional line of intersection. That is, finding points of
intersection leads to five equations with six unknowns, which gives one
degree of freedom.
So, if your "planes" intersect, what happens to your notion of the
distance between them? I am sure you have something else in mind for your
concept of distance but I have no idea what it is. It is as though you
asked for the distance between two non-parallel lines in two-dimensional
space. In general the lines must intersect at some point.
If you had selected only two principal eigenvectors, then your request
might make a certain kind of sense. It would be analogous to asking for
the closest distance between two one-dimensional lines in
three-dimensional space. In general such lines will not intersect and do
have a well-defined distance of closest approach. However, in your
five-dimensional space, even here, if your two pairs of selected
eigenvectors come close to being linearly dependent, such a closest
approach may occur at points that are very far away from the original
defining sets and therefore possess no particular significance with
respect to any reasonable notion of "similarity" between them.
I rather think that your notion of similarity must necessarily involve a
comparison between each of the three principal eigenvectors of the two
sets as well as their two mean value vectors. Such a similarity is not a
simple one-dimensional concept that can be expressed by a single numerical
quantity, in my opinion.
Roger Stafford
.
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