Should correlation coefficients be expressed as angles?
Edit 11/28: Edited note at bottom to note that the random variables should have finite variance, and that this is essentially just L². Also some formatting changes. This is something that has been bugging me for a while. The correlation coefficient between two random variables can be interpreted as the cosine of the angle between them[0]. The higher the correlation, the more "in the same direction" they are. A correlation coefficient of one means they point in exactly the same direction, while -1 means they point in exactly opposite directions. More generally, a positive correlation coefficient means the two random variables make an acute angle, while a negative correlation means they make an obtuse angle. A correlation coefficient of zero means that they are quite literally orthogonal. Everything I have said above is completely standard. So why aren't correlation coefficients commonly expressed as angles instead of as their cosines? It seems to me that this would make them more intuitive to process. Certainly it would make various statements about them more intuitive. For instance "Even if A is positive correlated with B and B is positively correlated with C, A might be negatively correlated with C." This sounds counterintuitive, until you rephrase it as "Even if A makes an acute angle with B and B makes an acute angle with C, A might make an obtuse angle with C." Similarly, the geometric viewpoint makes it easier to make observations like "If A and B have correlation exceeding 1/√2 and so do B and C, then A and C are positively correlated" -- because this is just the statement that if A and B make an angle of less than 45° and so do B and C, then A and C make an angle of less than 90°. Now when further processing is to be done with the correlation coefficients, one wants to leave them as correlation coefficients, rather than take their inverse cosines just to have to take their cosines again later. (I don't know that the angles you get this way are ac
Yes, thanks; I think a thing worth noting here, that is a reason I originally wrote this in the first place, is that people often don't speak in a way that strongly distinguishes "this state, instantaneously" and "this system, extended in time". Thus I think a lot of the reason that 2/3 cause such confusion is that the debate often looks like Alice saying "Oh, we'll just put a a spacecraft at L1 to deal with that", Bob replying "But that isn't possible", and Alice being confused and going "What? Of course you can put a spacecraft at L1". Yes, you can put a spacecraft at L1, but it can't... (read more)