Let me start by saying that I have decided to take a holiday this weekend… for me and my American readers, this is Thanksgiving weekend. I have decided that
there will be no technical post on this coming Monday, November 29.
While I expect to be doing mathematics this weekend… it is, after all, my idea of fun… I will treat myself to not having to turn some mathematics into a post this weekend.
I’ve spent some time since last Saturday’s post looking through other data sets available on the Internet. So far, I have found nothing as good as Ramanathan’s data, at least for illustrating stepwise regression and backward selection. While I hesitate to use a third data set from his website, I really haven’t found anything else as good, for what I want to demonstrate.
As for other mathematics… I’ve been nibbling at the edges of aircraft control, abstract algebra, the Kalman filter, and computer performance modeling. Nibbling, I say… getting somewhere, I’m sure… but I haven’t passed any milestones.
So let me tell a story… about final exams, and the stellar interior equations.
I took two final exams at Caltech that I thoroughly enjoyed, one as a sophomore and one as a junior. Both of them had the same format: the first part consisted of a lot of questions for each of which we were supposed to provide a one or two sentence answer; the second part asked us to pick something from the subject and go to town on it.
That is, part one asked us to demonstrate a broad familiarity with the subject… and part two asked us to demonstrate detailed familiarity with some specific part of it, of our choosing more or less.
The two subjects were… nonparametric statistics and astronomy. As I recall, but I could be wrong, the statistics final gave me a selection to choose from, while the astronomy final gave me a list of suggestions, but was open-ended in principle.
The usual first course in statistics is not called “parametric statistics”, except in contrast with “nonparametric statistics”. Roughly speaking, “parametric statistics” teaches us how to make inferences about the probability distribution of a population… given a sample drawn from that population. In particular, we attempt to infer the parameters of the probability distribution of the population, hence the term “parametric”.
“Nonparametric statistics”, on the other hand, is a collection of unrelated techniques for when we have even less information than usual.
One such technique, which I might elaborate on some other day, is to count the number of inversions in a sequence of integers… such as the draft numbers from the US draft lottery for 1970.
(The sequence 3,2,1 has 3 inversions: 3 precedes both 2 and 1, while 2 precedes only 1. Altogether, there are 6 possible orders, and they have a total of 9 inversions. The average number of inversions for the 6 orders is 9/6 = 1.5; and the population variance is 11/12. Now do that for the integers from 1 to 366. And then compare it to the actual number of inversions in the actual lottery.)
While I was supposed to be studying for my final exam in nonparametric statistics (and I needed to do well), I got distracted by trying to generalize a one-dimensional test (FYI, Wald’s Sequential Probability Ratio Test) to higher dimensions… I kept telling myself that I should be studying the material we had covered, but I just couldn’t stop playing with the generalization. I was having too much fun to worry about my grade.
Lo and behold, the final exam gave me the opportunity to present my generalization! It asked me to generalize the test to two dimensions, and I started out with, “It’s just as easy to generalize this to N dimensions….” It was wonderful… all the time I had spent playing with something interesting turned out to be A Good Thing.
For the record, I agree with Mark Twain: I have [tried, before I left school, if not all through it, to] never let my schooling interfere with my education.
For the record, I agree with whoever said: the mind is not a vessel to be filled, but a torch to be lit.
I think those two final exams were an interesting compromise between filling my mind and lighting it up. And it says something that even today I remember what topics I chose to expound on.
I did not get similarly distracted while studying for the astronomy final… but, as I recall, part two was open-ended: I could pick anything I wanted, and try to explain it.
I chose the stellar interior equations. They were far and away the most fascinating thing in the book… I had loved astronomy as a little kid, but now that I was a big kid, I had calculus at my fingertips, and I could write about differential equations. The stellar interior equations had simply not been accessible to me before college.
All I had to do then was explain the differential equations… closed-book, but that was no real problem. Okay, one of the equations was tough, and still is… but what I look forward to today — not literally today — is solving them numerically for the sun.
Hmm. It could happen today….