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Decius comments on Should correlation coefficients be expressed as angles? - Less Wrong
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Comments (24)
Yes. If you have n random variables, you can restrict to the subspace spanned by them (or their equivalence classes). For instance, if we have 3 random variables whose equivalence classes are linearly independent, then the span of these equivalence classes will be a 3-dimensional real inner product space, which will then necessarily be isomorphic to R^3 with the dot product.
Actually, it occurs to me that I've never actually sat down and checked that angle addition obeys the triangle inequality in 3 dimensions -- I suppose if nothing else it can be done by lots of inequality grinding -- but that's not relevant. The point is, it does hold in dimension 3, and hence by the argument above, it holds regardless of dimension, finite or infinite.
That's half of it- does there exist any set of angles which are mutually compatible angles on the n-dimensional surface but not inverse cosines of correlations?
No. Given any mutually compatible angles (which means we can choose unit vectors that have those angles) we can generate appropriately correlated Gaussian variables as follows: take these unit vectors, generate an n-dimensional Gaussian, and then take its dot product with each of the unit vectors.
Now the hard question: Is there a finite number n such that all finite combinations of possible correlations can be described in n-dimensional space as mutually compatible angles?
My gut says no, n+1 uncorrelated variables would require n+1 right angles, which appears to require n+1 dimensions. I'm only about 40% sure that that line of thought leads directly to a proof of the question I tried to ask.
Your gut is right, both about the answer and about its proof (n+1 nonzero vectors, all at right angles to each other, always span an n+1-dimensional space). You should trust it more!
I think that my 40% confidence basis for the very specific claim is proper. Typically I am wrong about three times out of five when I reach beyond my knowledge to this degree.
I was hoping that there would be some property true of 11-dimensional space (or whatever the current physics math indicates the dimensionality of meatspace is) that allows an arbitrary number of fields to fit.