I can’t say that it’s been a slow week for mathematics – but it hasn’t been very productive either.

I did write up one of the dozen or so small things for which I’ve done the mathematics; namely, Andrews curves. They were mentioned by Jolliffe as a possible tool for deciding how many principal components to keep in a PCA analysis. I couldn’t figure out what they were from his description. But I’ve known for a while now, and I finally wrote them up.

I do hope, however, that I don’t have to publish that post this weekend. I really would like to build a reserve. But we’ll see what happens.

The real challenge is that I am torn between 3 possible posts… 3 possible data sets… for the next multi-collinearity post. If I can’t settle on one to work on, I won’t get any of them finished. So we’ll see what happens.

I’ve made a little more progress in Stillwell’s “Naive Lie Theory”.

In the meantime, I’ve been browsing. In somewhat specific attempts, I’ve been looking at single deletion statistics again, taking notes on discussions of the care and feeding of multi-collinearity: detection, isolation, severity, and how to cope with it. I’ve also been looking at more general regression estimators – ridge regression and Bayesian methods –and at the Box-Cox family of transformations of the dependent variable for minimizing the error sum of squares.

But that was either taking notes or just reading stuff – oh, I worked out one, just one, transformation.

More generally, I spent an evening flipping through all of my books on commutative algebra and algebraic geometry while watching reruns of Criminal Minds – it’s one of my smaller collections of books, only a dozen in fact. Here’s what I’ve learned (!):

Commutative algebra is essentially the study of commutative rings. Roughly speaking, it has developed from 2 sources: (1) algebraic geometry and (2) algebraic number theory. In (1) the prototype of the rings studied is the ring of polynomials k[x1, …, xn] in several variables over a field k; in (2) it is the ring Z of rational integers.

(Atiyah & MacDonald, “Introduction to Commutative Algebra”, 0–201–40751–5, p. vii.)

Oh, I’ve also learned that I won’t be starting with that book; I will probably use volume 1 of Zariski & Samuel, “Commutative Algebra”. My copy is old, 1958, and predates ISBN numbers. A quick check of Amazon shows only volume 2 available.

When I pick up algebraic geometry proper, I will resume with Cox, Little, & O’Shea’s “Ideals, Varieties, and Algorithms”, ISBN 0-387-97847-X.

In any case, however, I probably won’t get to this anytime soon.