Rather than wait until I have a finished piece of math to put out here, let me do what I proposed: talk about the doing before the done.
I’m still working on principal component analysis (PCA). Roughly speaking, that’s a collection of techniques whose genesis was computing the eigenstructure of a correlation or covariance matrix. I’ve been working on it way longer than I expected to.
It’s been fun. Oh, I feel like I’m learning the way the Babylonians did: here are 5 examples; you figure out the general case. It’s been a puzzle different from what I usually do when I study something new.
Right now I’m trying to itemize the different… “conclusions” isn’t the word I want. Ah, “results” is closer, not quite exact, but I’ll go with it. I’m trying to list the different results that people present. It’s been a month since I sorted out the different ways of preprocessing the data, and I’ve been struggling with some of – ok, just about all of – the results people get.
I’m using books from chemistry, oceanography, geology, social sciences, and statistics – and they all do different things. They don’t even all call it PCA, but that’s ok: the initial computations are fundamentally the same. (I’m not sure they all know that.)
Anyway, as of yesterday morning there were two kinds of graphs that confused me. I think I made sense of one of them last night. The first actual mathematics I do this morning will be to confirm that my guess is right, that they are graphing what I now think they are.
I had stopped working when I knew darned well that I was confused. I read fiction – fantasy, in fact. (Now that’s an interesting juxtaposition!) The point is I had stopped actively investigating and was instead leaving my mind free to search for some connection between what I was seeing and what I knew.
I think it worked, but I have to confirm the connection.
The other thing I started a couple of days ago – after the subject came up over lunch with a friend – was a list of “25 math things to do before I die”. Oh, they told me years ago to do that for my life in general (Instruction #435, “Life’s Little Instruction Book”; H. Jackson Brown; Rutledge Hill Press, 1991; ISBN 1-55853-102-5.), but I’ve pretty much just bounced from one piece of math to another as external chance and internal whim have moved me.
Work is one thing, but this is play. Still, I can’t do every amusement park in the world, so maybe I should remember which ones I really want to go to.
It seems harder to build such a list for mathematics than for my life. Maybe that’s simply because I’ve never put it in those terms before. Sometimes I look at a shelf of books and say, “I need to get back into that area.” Sometimes a single book catches my eye and I think, “I’ve been meaning to work thru that book.” Sometimes I look at a shelf and I remember, “I put that stuff down because I hit that problem. I need to work on that problem again.”
As I said, I never expected that PCA would take more than a week. But it’s moved from its social sciences beginnings out into the physical sciences, and people in different fields have extracted different imaginative, innovative results. Had I been thinking in terms of a long-term list, I should have stopped working on PCA as soon as I realized what a project it would be.
It’s also been a great way to study the Singular Value Decomposition (that’s the first piece of math I’m trying to write up). I don’t at all regret the time I’ve put into PCA. A long-term list notwithstanding, this has been a fine short-term project. Still, I’d like to move on to something else with the New Year.
Anyway, that long-term list needs a lot more work before it sees the light of day here.
Now, back to PCA.