(Edited 1 Oct to replace an equals sign by approximation. Find “Edit”.)
There are so many things I could talk about today but I’m not sure where to start or how many of them to cover.
So let me start at the beginning.
When I woke up this morning, I realized that I had answered a fairly long-standing question pertaining to multicollinearity. (Yes, I wake up dreaming about math.) Assume for the moment that we have exact linear dependence among the columns of a matrix X. That is, we have an equation
X C = 0.
Note that we can multiply that equation by any scalar a, so we really have
Assume further that we have fitted the equation
We could combine them as
In other words, even if we could find the beta vector, we could add any multiple of the C vector to it.
In contrast, if we had multicollinearity instead of linear dependence, then we would have
X C ~ 0,
and for small values of alpha,
. (Edit: I had equality instead of approximately.) This in turn would give us
That in turn might be showing us why the computed coefficients can be sensitive to multi-collinearity: small changes in the data could lead us to a nonzero value of alpha, changing the vector beta by a multiple of the vector C.
Fine… now how do I find C?
And that is what I woke up knowing that I knew. I have been gleefully finding C for months – it’s the rightmost column of v in the singular value decomposition of X
X = u w v’.
I have been computing X.v and seeing how close to zero its rightmost columns were.
I don’t know if I’m going to be able to exhibit this behavior in an example… but I’m going to try.
And all that before I was even fully awake!
The early part of a “school day” morning is almost automatic: once I’m out of bed, I check the Internet while I wake up… mail here, mail there, my blog, newsgroups, and the non-research mathematics stack exchange (henceforth “mathematics stack”).
More and more, I am disappointed in the moderation of the mathematics stack.
Let’s see. When I first talked about it, I discussed a fascinating question: what one result in mathematics most surprised you?
If you were to go look at that question now, you would find that it has been closed to further discussion.
closed as too localized by t.b., Zev Chonoles. Sep 5 at 22:18
This question is unlikely to ever help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet.
If you were to go look at that question now, you would also find that it has 91 answers and a rating of 45.
I’ve seen damned few – if any! – questions out there with ratings of 45. I’m afraid the word “narrow” in the boilerplate justification is more applicable to the moderators than to the question.
Let’s see. Somebody asked a few days ago, “what the hell is a field?” That question has been closed too, and also includes the comment “no one says «what the hell is X?» here. Please be civil.”
Let me say that I can see a clear distinction between “what the hell is X?” and such alternatives as, “what the hell are you talking about?” or “who the hell are you to criticize me?”
I recall telling you about a middle school math teacher I know who was tutoring an adult in college algebra… they discovered the joy of swearing at mathematics.
I think swearing at mathematics is just dandy. I do it a lot. I’ve even done it a few times out here on my blog. (Mathematica’s inversion of a non-invertible matrix comes to mind.)
(I’ll freely confess that the question “what the hell is a field?” startles me. After all, the rational numbers are a field, and we learn about them pretty early on. On the other hand, the questioner was using Lang’s “algebra” – and Lang is not the easiest of authors to read.)
And those are just the tip of the iceberg. The FAQ says
Post good, relevant answers, and if some (but not all) happen to be about your product or website, so be it…. If a huge percentage of your posts include a mention of your product or website, you’re probably here for the wrong reasons.
And I’ve seen them enforce this, jumping on someone for giving admittedly good answers too many of which pointed to his site.
So. When someone asks how to find the ZYX Euler angle sequence for a rotation specified by a quaternion – I should not tell them that I have solved that problem on my blog.
But anyone else in the world could provide that link.
Anyone but me, because it’s my blog.
To say I’m disappointed would be an understatement. I get no benefit from increased traffic on this blog… but I hate seeing questions go unanswered. And I feel a little embarrassed complaining about this – but it really is a damned shame that the mathematics stack is so… narrow-minded.
I could, of course, simply go ahead anyway and post answers that lead to my blog. But I firmly believe in the dictum “my server, my rules”: they get to define what is spam, and I will honor their definition.
Alternatively, I could post a comment inviting the questioner to repost his question on the newsgroup sci.math, where I won’t be violating policy.
And all of that was before 8:30 AM. Well, I’ve been taking notes on the mathematics stack for a while… all I did this morning was read the latest questions. Yes, of course, I still read the mathematics stack!
Then I actually spent 3 hours chasing down some fascinating links… but I think I’ll tell this story next week.