Re: how to compute distance metrics with multi dimensional data

From: Lou Pecora (pecora_at_anvil.nrl.navy.mil)
Date: 02/11/05 

Date: Fri, 11 Feb 2005 08:55:31 -0500

In article <1108112208.194608.77050@l41g2000cwc.googlegroups.com>,
 "bluelagoon" <bluelagoontrading@hotmail.com> wrote:

> what you are saying that: to compute distances i need first to z-score
> s1,s2,s3 so that i can compare apples to apples?

Yes, assuming z-score means demean and standard deviation scaling. I am
not familiar with that term.

> ok, then what is the formula for computing euc. manhattan and max norm?
> i know how to compute for 1d data, i can't seem to find any reference
> paper on computing multi dimensional distances... do you have any
> points for me?

I think the max norm is just a matter of going through each component
and find the max abs value.

> thanks.
> ps. ok another example: suppose you have a time series with 3d
> coordinate vector (x,y,z) series, ie a location in 3d space.
> you need to embed with dim=4 and delay=1, ie you get a 4 point
> trajectory, then how do you compute distances between the embedded
> vectors ie with 3d embedded time series ?

Just treat the embedded vector for what it is, a 4D vector. So you
would compute the Euclidean norm for v=(x(t), y(t), z(t), x(t+1)) where
I assume you used the delay on x as,

  ||v||=sqrt( x^2(t)+ y^2(t)+ z^2(t)+ x^2(t+1) )

Pretty straightforward.

I have to say that your wording in the above paragraph leads me to
suspect that you are viewing the embedding process differently. Maybe
I'm wrong. If what I've written doesn't seem right, then we need to
take a step backward and look at the objects you are building that you
call 4 point trajectories, etc.

-- Lou Pecora (my views are my own)