Conference rusure::math

148.0. "Most Likely Vector" by HARE::STAN () Sun Sep 16 1984 15:36

Newsgroups: net.math
Path: decwrl!decvax!cwruecmp!ryan
Subject: curve fitting (maybe)
Posted: Fri Sep 14 09:24:42 1984

Problem
-------

given: any number of points in n-space and any number of vectors (of course
   	also in n-space) associated with each point.

find:  for any given point, x, find its most likely vector.


    eg 	|    .------>     .
	|     .x         /|
  (n=2) |       .-->    / |
	|    ^         /  |
	|    |        L   v		x's vector would probably turn out to
	|    |      			    be something like  .---->
	|    |        <---.
	|    .   <-------.
	|       	/
	|              /
	|	      L 
	|_________________________  

I'm not really sure what I mean by 'most likely'.  I know I don't mean straight
averaging so that every point's 'most likely' vector would be the same.  
A weighted average might be the way to go, with the closeness of x to the
other points as the weighting factor.

If it'll be any help, in the exact problem that I plan on using the solution
in both the points' locations and the vectors will be integers.  (ie Point
(1,5,2,3,2) can have vector (7,-5,1,0,2) associated with it but not the
vector (pi,.5,e,i,sqrt(2)). Also point (.5,0,0,0,-.5) won't have any vectors.)
If you want, x's most likely vector can be made of either integers or reals.

One last note: I don't keep up math that much, so if it has been proven that
	       this problem is unsolvable, I'm sorry I wasted your time.

If anyone gets the solution or would like me to send them a copy of it (when
and if I get it) mail me (don't post to the net).

						Ryan McGuire

From:	ROLL::USENET       "USENET Newsgroup Distributor" 16-SEP-1984 22:01
To:	HARE::STAN
Subj:	USENET net.math newsgroup articles

Newsgroups: net.math
Path: decwrl!flairvax!turtlevax!ken
Subject: Re: curve fitting (maybe)
Posted: Sun Sep 16 11:56:33 1984


> given: any number of points in n-space and any number of vectors (of course
>    	also in n-space) associated with each point.
> 
> find:  for any given point, x, find its most likely vector.
> 

Sounds like a quantization problem to me.  There was a paper in maybe
IEEE computer graphics and applications recently about scanning a sample
space with Peano curves to get clusters of sample points.

If your quantization values ("vectors") are fixed, then all you need is
an appropriate metric to determine the distance of each "point" to each
"vector".  Then pick the closest "vector".

I've been using quotes because of the nonstandard usage of the term "vector".
A vector is a distance with a direction, and has no root; i.e. it can float
around all over the place.  I suspect that you really mean a vector rooted
at the origin.  That brings up the question as to why you didn't use the term
"point" for both.
-- 
Ken Turkowski @ CADLINC, Palo Alto, CA
UUCP: {amd,decwrl,dual,flairvax,nsc}!turtlevax!ken
ARPA: [email protected]