From: gbnewby@uxh.cso.uiuc.edu
Subject: Re: Sources of the concept of "information as space"? 
Date: Thu, 09 Jan 92 10:58:27 CST
Organization: University of Illinois


(Hi, Chris!  Long time no hear.)

I wanted to add a few things about Chris' pre-vacation comments on
the non-euclidean nature of perceptual space.  This is a fascinating area,
IMHO, but not a necessary part of mainstream VR...

I'll include a few references and a quick rundown of how such psychometric
data are collected, too.

>object-object similarity is often regarded as some
>sort of inverse to distance.  The greater the similarity between objects is
>judged to be, the "closer" we say they are.
>
>One of the empirical results from the research I and others did at New Mexico
>State some years ago was that if you examined people's pairwise similarity
>judgements for a set of objects and converted the similarities into distances,
>you found widespread violations of the metric axioms that define the nature of
>euclidean space.  The axioms violated most often were symmetry
>(distance(A->B) == distance(B->A)) and the triangle inequality, although there
>were a smaller number of cases where the "distance" from an object to itself
>could reasonably be interpreted as nonzero, which violated the third axiom.
>Examples of symmetry violations are:
>
>	1) A woman was judged to be more like a rose than a rose like a woman
>	2) the word "orange" is more like "fruit" than "fruit" is like "orange"

In the psychometric method of multidimensional scaling (MDS), assymetry
is a well-known but usually ignored phenomena.  There's an excellent
introduction to the topic published by Sage (one of their small green 
books on Quantitative Applications in the Social Sciences, sorry I
don't remember the exact title or author).  

The Galileo method for MDS, with which I am intimately familiar, is one
of few methods that take the non-euclidean aspects of PERCEIVED space
into account.  

Chris mentioned assymetry of class-object relationships.  Other types
of assymetry have also been found (causal or time-based relationships,
for instance).

Another aspect Chris didn't explicitly mention is that of the non-euclidean
nature of conceptual space in the first place.  That is, if you give
people a bunch of paired-comparisons to make, the outcome usually
can NOT be plotted in a 'real' (euclidean) space.  Galileo methodology
uses a Riemann space (which includes 'imaginary' dimensions).  This
is the rule, not the exception.

Most MDS methods ignore these things completely, and either make
a best-fit solution in a euclidean space, or (more likely), simply
solve for the first 3 or so dimensions anyway.  Galileo solves
the ENTIRE space -- this means that the resulting Riemann space
represents faithfully ALL the paired-comparison relations.  Of
course, you might not need all that precision...

Two people who have written extensively in related areas:

	Philip Johnson-Laird has written about spatial perception
		and spatial encoding for human memory

	Eleanor Rosch developed 'prototype theory,' which 
		describes and explaines the types of object-
		class relationships Chris mentioned.  For instance,
		how a kitchen chair is more like a prototypical
		chair than a dentist's chair.

Nuff said -- what follows is a citation and a real quick rundown of
how MDS data are collected, if anyone read this far and is still
wondering.

For Galileo:  Woefel, J.D. and E.L. Fink (1981).  The Measurement of 
	Communication Processes:  Galileo Theory and Method.  New
	York:  Academic Press.  

To get the data to build a space, you first choose a ruler.  For
instance, 

	"Good and Bad are 100 units apart"

(space, dissimilarity, and distance are all concepts which are measured
with a subset of themselves -- as is time).

Then, identify the concepts which you want to measure (your ruler
concepts should probably be included).  For instance, 
	
	good, bad, stressful, tired, healthy

(concepts are usually part of the same conceptual 'domain,' but they
don't have to be)

Finally, measure pairwise relationships, using your ruler.  

	If good and bad are 100 units apart, how different or
	far apart are:

		good and stressful
		good and tired
		good and healthy
		bad and stressful
		bad and tired
		....etc...

As you can see, the number of comparisons grows quickly with the number
of concepts.  And, you want to collect data from a large sample 
(usually) and take the average for the next step:  building the
space.  

I won't cover the specifics here on building the space -- there
are many different methods.  Basically, though, you extract
principal components (aka the axis' or dimensions for the space)
from the data -- this is conceptually similar to taking a regression
line from multivariate data.  Voila!  You place all the concepts 
in the space.  Then, you can visualize the space  (at least the 
first 3 dimensions) or watch how concepts move over time, 
by collecting more data, or whatever.

-- Greg Newby

Grad Sch of Lib and Info Sci / Nat'l Cntr for Supercomputing Apps
U of Illinois at Urbana-Champaign
gbnewby@uxh.cso.uiuc.edu