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
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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
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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
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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.

## Comments (24)

BestThat's half of it- does there exist any set of angles which

aremutually compatible angles on the n-dimensional surface butnotinverse 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.