Re: Angles between vectors

From: David Jones (dajxxx_at_ceh.ac.uk)
Date: 10/22/04 

Date: Fri, 22 Oct 2004 10:05:18 +0100

Ross Clement wrote:
> So, while I'm thinking of classification, I'm thinking about
> classification after the vectors describing the books have been
> reduced (by some method) to a single dimension. And no cheating like
> making that a complex number or using bit patterns to include
several
> numbers. There are a lot of ways that vectors can be reduced to a
> single dimension (distances from the origin, angle from the origin,
> principle component, etc), and as a very vague conjecture, I thought
> that angles from some origin might be the best method. Not that it's
> likely to work, but I thought it would be interesting to think about
> it.
>

A question that comes to mind is whether you are trying to go too far
in reducing the number of dimensions. Going from 2 to 1 dimension is
OK, but should you be thinking first of 3 to 2, rather than 3 to 1.
The obvious analogue of your 2 to 1 (giving angle, or possibly better
the distribution on a cicle) is to reduce 3 dimensions to a
distribution on the surface of a sphere. Thus you could thinj of
forming clusters on the surface of a sphere. More generally you could
be thinking of distributions on hyperspheres. There are books on
distributions of directions (directional data, circular data, angular
data) that may contain leads on how to think of multidimensional
cases. One book is by Mardia (1972, Statistics of Directional Data,
Wiley) but there are several more recent books.

David Jones

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