It’s been an uneventful week mathematically.
First a blog note: last Monday’s post was number 300.
My alter ego the kid came across one interesting piece of mathematics, while looking at something called Independent Component Analysis (ICA). The problem which ICA attempts to solve is to break apart a composite signal. Suppose you have 3 microphones in a room and 3 people speaking, and that each microphone picks up a weighted linear combination if the 3 people.
Can we figure out what the 3 separate speech records are?
I have barely looked at this, but along the way here’s what I saw…. We recall that two events are independent if and only if their joint probability is the product of the individual probabilities. More generally, two random variables are (stochastically) independent iff their joint probability distribution is the product of the two individual distributions.
But this means that if two random variables are stochastically independent, then their joint moments are the products of the individual moments.
Suppose we have zero-mean random variables x and y. If x and y are independent, then for any integers p and q, we have this equation for the expected values of powers:
E(x^p y^q] = E(x^p) E(y^q).
For p = q =1 that says that the mean of the product is the product of the means: E(x y) = E(x) E(y).
For p = q = 2, that says that E(x^2 y^2) = E(x^2) E(y^2)… and so on.
But it also gives us equations for p ≠ q, like E(x y^2) = E(x) E(y^2).
I emphasize that this is probabilistic independence, not linear independence. For example, if I have column of data, x, and I form a column x^2 in order to fit a quadratic term… x and x^2 are linearly independent but not stochastically independent.
Now, I have no idea if this equation can be useful… but I expect I’ll be playing with it.
Finally, I received that print-on-demand thing entitled “Multicollinearity” that I mentioned a few weeks ago. I had no idea whether it was very good or very bad.
My worst fears were not met.
They were exceeded. It wasn’t crap, but it was worthless.
Good math but of no value? How could that be?
What I got for my $35 plus shipping was 13 articles, 57 pages, from Wikipedia… like I really need a hardcopy of material freely available to me at the click of a mouse. And I do have a printer of my own!
Worse, the only article I read from the “book”, the 3-pages (!) on multicollinearity, wasn’t even up to date: the current Wiki article has a little more material.
Still, as lessons go, this was cheap. I knew I was taking a chance; there was, after all, no description of what it was I was ordering. Now I know why.
I can’t imagine that I will ever order another product of “Alphascript Publishing”; nor another product from the bookstore that actually sold this.
Fool me once, shame on you; fool me twice, shame on me.