I decided to play hooky from school yesterday, so here’s the homework that was due yesterday.
By around 10 AM, I had a fairly simple happenings post laid out in my head.
3 books came in during the week. Lorenz’ “The Essence of Chaos” begins fairly slowly, but I think it will be a worthwhile read. Mumford’s “The Red Book of Varieties and Schemes” looks excellent… it has a list of prerequisites which I do not have, but which tell me what I need to get out of my current study of rings… he says in the preface that “The weakness of these notes is what had originally driven me to undertake the bigger project: there is no real theorem in them!” Instead, “The hope was to make the basic objects of algebraic geometry as familiar to the reader as the basic objects of differential geometry and topology….”
Sounds like a valuable resource.
Bloomfield’s “Fourier Analysis of Time Series” is fascinating. It’s fairly small, 250 pages… it may only have 3 significant examples… but I’ve already persuaded Mathematica to work out one of them – something other than the straightforward computation of the discrete Fourier transform of the data, followed by the periodogram. Instead, I did a nonlinear model which fitted frequencies directly. I’m sorry it’s taken me so long to get around to that.
I spent Friday evening curled up with that book. (The data for it, however, is rather challenging to track down. So far I have found both datasets I was looking for – but neither is any longer where the book said they could be found.)
And I’ve ordered 2 more books – both pertaining to perspective and cameras.
And that’s what I was ready to write yesterday morning. It would have been a simple straight-forward post.
Ho hum. I decided to take a look at Peter Woit’s blog to see what was happening in particle physics. Was there anything exciting I could talk about?
The signal from the Large Hadron Collider at 125 GeV still holds up… and it might be the Higgs particle… but there is no sign of SUSY particles – that is, the supersymmetric fermions and bosons corresponding to known bosons and fermions respectively – where some were expected… and he had an interesting link which I followed.
For quite a while. The question was “What is your favorite deep, elegant, or beautiful explanation” … and there are almost two hundred answers, from a wide variety of people (Alan Alda, among them).
And after all that, I just did not feel like doing what I was supposed to do – write the Happenings post, and do rings, and work on Monday’s group theory post.
Instead I decided to move beyond Friday evening’s reading. I curled up in my recliner with a Bayesian statistics book, Phil Gregory’s “Bayesian Logical Data Analysis for the Physical Sciences”. (He uses Bayesian methods for finding extra-solar planets, i.e. around other stars.)
One of my open subjects is spectrum analysis – frequency domain analysis – of time series. That’s why I bought and browsed the Bloomfield book that came in during the week. One of my other books categorized methods as “classical” or “modern”… classical being smoothing the periodogram, and modern being comparing the given spectrum to the spectra of explicit time series models. (I’m sure that’s too simple a summary, but sometimes simple is appropriate.)
The periodogram is just the magnitudes of the Fourier coefficients; it needs to be smoothed because it’s often violently erratic. And smeared out, too. (“Bin leakage”, if the number of observations is not an exact multiple of the periods in the data… which can be rather hard – usually impossible! – to arrange.)
I’m inclined to add “post-modern”, meaning Bayesian.
The simple form of Bayes Theorem is
p(A|B) = p(A) p(B|A) / p(B),
where p(A) and p(B) are the probabilities of A and B, and p(A|B) is the (conditional) probability of A given that B has occurred. Similarly for p(B|A).
The proof is easy enough:
p(A) p(B|A) = p(A B) = p(B A) = p(B) p(A|B),
where p(A B) is the probability of both A and B happening. It’s the same as p(B A), and each of those can be written using a conditional probability. Equate the two expressions involving conditional probabilities, and then solve for p(A|B). Voila’, Bayes Theorem.
Seems harmless enough, huh?
Well, for one thing, it flies in the face of the far-more-common frequentist structure of probability and statistics… and, in the case of spectrum analysis, it is significantly more precise than classical methods. So much more precise that people mistrust it. As I said, it’s being used to find extra-solar planets.
One of the books that popularized it in the physical sciences is now available online, having gone out of print. (Im very glad I found it online; the only copy – the only copy! – available through Abebooks was in Berkeley, and I was thinking about driving up to get it. Instead, I downloaded it.
The second book that really grabbed me was Sivia’s “Data Analysis: A Bayesian Tutorial”. It begins with an awesome example: computing the probability that a coin is biased as we throw it more and more times. I’ve been meaning to work that example for the blog.
The first book that really grabbed me was Tribus’ “Rational Descriptions, Decisions, and Designs”, a long time ago. (Good luck finding that! My copy was used when I bought it 35 years ago.)
OK… now let me answer this question about the quaternion group and permutations….