[R] distance coefficient for amatrix with ngative valus

[Rolf Turner]

On 04/10/11 17:05, R. Michael Weylandt wrote:

<SNIP>
> More importantly, as I said in my initial response, any distance
> metric worth its salt is translation invariant.

<SNIP>

Point of order, Mr. Chairman.  (This is really *toadally* off topic;
my apologies, but I couldn't resist --- I trained as a pure mathematician).

A *metric* need not in general be translation invariant.  Indeed a metric
need not be defined on a space in which translation makes any sense.

A metric defined in terms of a *norm* (on a normed vector space)  by
rho(x,y) = ||x - y|| is of course by definition translation invariant, 
and that's
what most of us think in terms of.

But there are perfectly ``reasonable''  metrics, defined on vector spaces,
which are not translation invariant.  Whether these are ``worth their salt''
is I suppose a matter of taste.  (You should pardon the expression. :-) )

A simple e.g. of a non-translation-invariant metric is

     rho(x,y) = |x - y|/(1 + |x| + |y|)

(defined on the real line).  It is easily checked that rho(.,.) 
satisfies the
four conditions that a metric must satisfy.  (Exercise for the interested
reader.)

Note that rho(1,2) = 1/4  but rho(2,3) = 1/6, ergo not translation 
invariant.

     cheers,

         Rolf Turner