Eigenvectors and Factor Analysis

Now here is the point where I have to drag in a little bit of real math. We associate the complex spectrum of a linear device with a fourier transform, in which a pair of numbers is given for each frequency. But strictly speaking the fourier transform is no more and no less than a scheme for representing a mathematical function in terms of sine and cosine waves. (A cosine wave is just a sine wave phase-shifted 90 degrees.)

These wave forms are just a convenient set of eigenvectors or coordinate axes, which just happen to be useful for electrical signals because of Maxwell’s equations. But any spectrum can be represented more or less well by any set of orthogonal functions, and that means we can look around for some functions more appropriate for our data.

A good example of this goes back a couple of years to some researchers who used factor analysis to come up with a good set of eigenvectors for comparing pictures of human faces. The principal eigenvector or best-single- coordinate-axis is simply an average human face, totally unmemorable, just the most typical human face, the average face.

Using that axis alone you have a single number which only says how typical a face is; a large positive number would describe a very ordinary face, and a large negative number an unusual face. Other coordinate axes are other eigenvectors, some of which describe squashed, twisted, or other orthogonal forms of extreme ugliness — oops, I mean extremely distinctiveness in character.

Since these coordinate axes are facelike, you can look at them, seeing what I recognize as archetypical faces — indeed when Jung was writing about archetypes I see him as writing about eigenvectors. And, of course, the archetypes Jung was most concerned about were archetypes of personality, and the Myers-Briggs personality types are derived from Jung’s work.

In psychology the use of factor analysis, (which is really just the extraction of eigenvectors from autocorrelation matrices generated from personality test scores) is very old news, having been used for decades.

But here is what is wrong about factor analysis and what is why we have so many different theories of personality instead of just a useful toolkit. Almost all psychologists using factor analysis have viewed the extracted eigenvectors as somehow real, like the ideals of Platonic philosophy which Plato considered as much more real that anything in our world, the world of illusion.

But when computer science people use the extraction of eigenvectors it is not because of any underlying reality attributed to them, but simply as a form of data-compression. Both the JPEG and MPEG data compression standards, used for compressing still images and moving pictures (respectively), make use of the extraction of eigenvectors.

The basic idea can be seen in the use of archetypical faces: instead of having to produce an image which might contain millions of pixels to describe a face as a picture, you simply compare any face you want to describe with the archetypical faces (which are eigenvectors of the matrix etc. etc) and since pictures of real faces were used in generating the matrix, the face you want to describe will resemble more-or-less (or differ more-or-less from) the archetypical faces.

You can say your face is a 64% match with one archetype, a 32% match with another, and so on, so that you can produce a very accurate description of a human face by using only half a dozen or so numbers, instead of the millions of pixel values otherwise required. You can compress the millions of pixels to very few numbers, then at any time your can reconstruct the original image with very little loss of detail.

But no computer scientist would go on record as thinking that the archetypical faces are more real than our own (tragically flawed) visages. No one would say God used those archetypes when designing the face of Adam or Eve.

Similarly, I say the INTP or ESFJ or any other personality description does not relate to any underlying Platonic reality — they are just data compression, to be used when useful. Oh, all right, sure, there are a few Platonists in the computer science community — we have just about the same number of metaphysicians and lunatics as any other community. But my point really is that eigenvectors or archetypes are not part of Science, they are part of Technology. (Forgive the capital letters, they are part of the Platonic Archetype of this argument, so my hands are tied.)

Copyright © 1998 Douglas P. Wilson

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