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Decius comments on Should correlation coefficients be expressed as angles? - Less Wrong Discussion

43 Post author: Sniffnoy 28 November 2012 12:05AM

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Comment author: Decius 28 November 2012 05:27:42PM 4 points [-]

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?

Comment author: Kindly 28 November 2012 06:04:42PM 8 points [-]

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.

Comment author: Decius 29 November 2012 05:24:04PM 1 point [-]

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.

Comment author: Kindly 29 November 2012 05:38:15PM 2 points [-]

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!

Comment author: Decius 30 November 2012 01:52:26AM 0 points [-]

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.