Logical but Loose (fuzzy) Euclidean Distances

Tue, 26 Mar 1996 14:02:54 -0500

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Bob and Alastair,
The Slater loadings in Circumgids were
derived from Slater's own equations.
We adjusted the loadings so that they
could be printed on our axis plots.They
give the same picture as Ingrid. You may
also analyze the rows and constructs
separately using Circumgrids. I seem to
recall from the 1970's that Ingrid
would do that too- and a heck of a lot more.
Its been a long time. By the way, Finn
Tschudi's package Flexigrid also handles
elements and constructs at the same time.
I seem to recall that he used factor scores.
Flexigrid is an interesting package. Do
ya'll know if there is a PC version of Ingrid?
Concerning the raw-score factor analysis.
The details are in Nunnally's text
Psychometric Theory. It is the cross
products of the Euclidean distances that
are factored. Nunnally defines these cross
products as the sum of products
across some context. For example, =

person a per. b cross product
var1 1.5 1.0 1.5
var2 .5 2.0 1.0
var3 -2.0 -1.0 2.0
var4 1.2 - .5 -.6 =

The matrix of such sums is the =

cross-products matrix. =

The distances are based on the
Pythagorean theorem, i.e. on the square
root of the sum of the squared distances. =

This was the metric used in the semantic
differential. When dealing with rating scales,
the factors of the correlations may differ
from the factors of the Euclidean distances.
This is because the Euclidean distances take
the actual difference between scores
into account, rather than just the similarities of
standardized patterns. Thus a grid containing
the following patterns would lead to different
factors, dependening on which metric you factored: =

(factor by rows)
4 6 8 2 3 5 =

9 1 3 7 8 0
5 5 7 3 4 4
Things can be positively correlated but
separated by large distances-
or negatively correlated and separated by =

small distances. To the extent that
ratings are used, there is a strong case for
looking at both correlations and
Euclidean distances. This is not needed,
however, when factoring ranks. The
factors will be the same, since the ranks are
already on a common scale.
Correlations implicitly put the Eucliden =

distances on a common scale by
standardizing them- thus removing potentially
important information. However, the ranks of
correlations would not necessaily equal the
ranks of the Euclidean cross products. =

In a sense Bieri's cognitive complexity
measure is a "poor mans" raw score
factor analysis. He focuses on absolute
differences of zero or 1, ignoring
other distances.
This brings up Alastairs points on fuzzy
sets. For Kelly, loosness was not the
same as lack of structure. Loose essentially
meant not precise, suggesting Alastair is
correct in approaching things by fuzzy set
theory. The danger is that his readers may
ignore his caveat that loose only "MAY" imply
lack of structure. This is the problem that
plagued the Bannister-Fransella Test. Low
intensity could imply either loosecomplexity
or fragmentation. Complex factor
structures were being confounded with low =

intensity (fragmentation). I do not know fuzzy
set theory but have pondered around it for
years. If one could show a variable relationship
between constructs in terms of low correlations
while using Euclidan distances to develop
similarity estimates, then convert the
Euclidean cross products into ranks before
passing them into a coordinate grid, one might
show that initial distances could be low in
correlation (loose, fuzzy) while being at all or
any level of similarity via Euclidean cross
products. AND be logical. This would allow =

for the assessment of loose but logical =

constructs. That might be relevant to creativity.
They may be rare critters but their possibility
should be recognized. It would be nice see that
Don and Fay were really on to something with
their intensity measure. They may have given
us some large shoulders to stand on.
Whatcha think?