Example: Is it a transition matrix? Part 2

We had three matrices from Jolliffe, P, V, and Q. They were allegedly a set of principal components P, a varimax rotation V of P, and a quartimin “oblique rotation” Q.

I’ll remind you that when they say “oblique rotation” they mean a general change-of-basis. A rotation preserves an orthonormal basis; a rotation cannot transform an orthonormal basis to a non-orthonormal basis, and that’s what they mean — a transformation from an orthonormal basis to a non-orthonormal basis, or possibly a transformation from a merely orthogonal basis to a non-orthogonal one. In either case, the transformation cannot be a rotation.

(It isn’t that complicated! If you change the lengths of basis vectors, it isn’t a rotation; if you change the angles between the basis vectors, it isn’t a rotation.)

Anyway, we showed in Part 1 that V and Q spanned the same 4D subspace of R^{10}\ .

Now, what about V and P? Let me recall them:
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Example: Is it a transition matrix? Part 1

This example comes from PCA / FA (principal component analysis, factor analysis), namely from Jolliffe (see the bibliography). But it illustrates some very nice linear algebra.

More precisely, the source of this example is:
Yule, W., Berger, M., Butler, S., Newham, V. and Tizard, J. (1969). The WPPSL: An empirical evaluation with a British sample. Brit. J. Educ. Psychol., 39, 1-13.

I have not been able to find the original paper. There is a problem here, and I do not know whether the problem lies in the original paper or in Jolliffe’s version of it. If anyone out there can let me know, I’d be grateful. (I will present 3 matrices, taken from Jolliffe; my question is, does the original paper contain the same 3 matrices?)

Like the previous post on this topic, this one is self-contained. In fact, it has almost nothing to do with PCA, and everything to do with finding — or failing to find! — a transition matrix relating two matrices.
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Books Added: Color 3

discussion

Having picked up color again, there are three books I want to add to the bibliography.

The authors are Kessler, Malacara, and Glassner.
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Transition matrix: to be or not to be

Cohen (“Visual Color & Color Mixture”, see the bibliography) did something very interesting. In fact, he did something useful which I had never seen before.

Although this post uses some matrices which we saw in the color posts, I think this can stand on its own: you need not have read the color posts. But if you are specifically interested in color, or in Cohen’s work, this post is very relevant.

He was trying to describe how to find a transition matrix between two given data matrices. This will come in handy — very handy! — whenever people give the alleged result of an unspecified linear transformation of a data matrix.
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Color: decomposing the RGB A transpose matrix

(abuse of) terminology

Sometimes I get tired of writing “xbar, ybar, zbar tables” — and I just write “xyz bar tables” or even “XYZ tables”. Similarly for rbar, gbar, bbar tables — rgb bar. I’m not talking about anything new, just abbreviating the names.

Introduction

This is the third post about the example on p. 160 of W&S. Once again, I am going to decompose the reflected spectrum into its fundamental and its residual.

This time, however, I’m going to use the rbar, gbar, bbar tables (RGB) instead of the xbar, ybar, zbar (XYZ) tables. They did not do this.

I’ll tell you now there is one little twist in these calculations. We will need the ybar table, because we still need to use it to scale our results.

In addition to showing, as I did previously, the dual basis spectra and the orthonormal basis spectra for the non-nullspace, I will display the orthonormal basis for the nullspace.
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Color: decomposing the A transpose matrix and the reflected spectrum

This is the second post about the example on p. 160 of W&S. Here’s the first. (Incidentally, I am going to decompose the reflected spectrum into its fundamental and its residual. I do not think that they did this. Oh, this is what I did for Cohen’s toy example, but now it’s for real. Approximate, but for real.)

review the XYZ for this

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