http://ordination.okstate.edu/PCA.htm
owever, with more than three dimensions, we usually need a little help. What PCA does is that it takes your cloud of data points, and rotates it such that the maximum variability is visible. Another way of saying this is that it identifies your most important gradients.
http://en.wikipedia.org/wiki/Mahalanobis_distance
In statistics, Mahalanobis distance is a distance measure introduced by P. C. Mahalanobis in 1936.[1] It is based on correlations between variables by which different patterns can be identified and analyzed. It is a useful way of determining similarity of an unknown sample set to a known one
http://onlinelibrary.wiley.com/doi/10.1002/wics.101/abstract
Principal component analysis
Hervé Abdi1,*, Lynne J. Williams2
http://www.cs.otago.ac.nz/cosc453/student_tutorials/principal_components.pdf
http://stats.stackexchange.com/questions/2691/making-sense-of-principal-component-analysis-eigenvectors-eigenvalues
I would never try to explain this to my grandmother, but if I had to
talk generally about dimension reduction techniques, I'd point to this
trivial projection example (not PCA). Suppose you have a Calder mobile
that is very complex. Some points in 3-d space close to each other,
others aren't. If we hung this mobile from the ceiling and shined light
on it from one angle, we get a projection onto a lower dimension plane
(a 2-d wall). Now, if this mobile is mainly wide in one direction, but
skinny in the other direction, we can rotate it to get projections that
differ in usefulness. Intuitively, a skinny shape in one dimension
projected on a wall is less useful - all the shadows overlap and don't
give us much information. However, if we rotate it so the light shines
on the wide side, we get a better picture of the reduced dimension data -
points are more spread out. This is often what we want. I think my
grandmother could understand that :-)
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