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From: saswss@hotellng.unx.sas.com (Warren Sarle)
Subject: Re: Multidimensional Scaling
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Date: Sat, 12 Nov 1994 20:10:20 GMT
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In article <Cz5u6G.159@acsu.buffalo.edu>, comjoew@ubvms.cc.buffalo.edu (Joseph D Woelfel) writes:
|>      Leon Heller suggests using Multidimensional Scaling as a representation
|> of cognitive structure. CATPAC provides MDS representations of its internal
|> states. Especially if the net is represented by a square matrix, the
|>      principle axes of the scalar products matrix fromed from this
|>      will yield coordinates for plotting each neuron in a MDS space;
|> the distances between neurons will represent their dissimilarities.

That's Torgerson's method invented around 1950. Although Tisdale and
Hecht-Nielsen might disagree, I would claim that there _has_ been
substantial progress in MDS methodolody in the last 44 years. :-)

Here are some references from the documentation for the MDS procedure
in SAS/STAT software, Technical Report P-229:

For an introduction to multidimensional scaling, see Kruskal and Wish
(1978) and Arabie, Carroll, and DeSarbo (1987).  A more advanced
treatment is given by Young (1987).  Many practical issues of data
collection and analysis are discussed in Schiffman, Reynolds, and Young
(1981).  The fundamentals of psychological measurement, including both
unidimensional and multidimensional scaling, are expounded by Torgerson
(1958).  Nonlinear least-squares estimation of MDS models is discussed
in Null and Sarle (1982).

References:

Arabie, P., Carroll, J.D., and DeSarbo, W.S. (1987), _Three-Way
Scaling and Clustering_, Sage University Paper series on Quantitative
Applications in the Social Sciences, 07-065. Beverly Hills and London:
Sage Publications.

Carroll, J.D. and Chang, J.J. (1970), "Analysis of Individual Differences in
Multidimensional Scaling via an N-way Generalization of the 'Eckart-Young'
Decomposition," _Psychometrika_, 35, 283-319.

Davison, M.L. (1983), _Multidimensional Scaling_, New York: John Wiley & Sons.

Heiser, W.J. (1981), _Unfolding Analysis of Proximity Data_, Leiden:
Department of Datatheory, University of Leiden.

Jacobowitz, D. (1975), _The Acquisition of Semantic Structures_, Doctoral
dissertation, University of North Carolina at Chapel Hill.

Krantz, D.H., Luce, R.D., Suppes, P., and Tversky, A. (1971),
_Foundations of Measurement_, New York: Academic Press.

Kruskal, J.B. and Wish, M. (1978), _Multidimensional Scaling_,
Sage University Paper series on Quantitative Applications in the Social
Sciences, 07-011. Beverly Hills and London: Sage Publications.

Null, C.H. and Sarle, W.S. (1982), "Multidimensional Scaling by Least
Squares" in SAS Institute Inc., _Proceedings of the Seventh Annual SAS
Users Group International Conference_, Cary, NC: SAS Institute Inc.

Rabinowitz, G. (1976), "A Procedure for Ordering Object Pairs Consistent with
the Multidimensional Unfolding Model," _Psychometrika_, 41, 349-373.

Ramsay, J.O. (1986), "The MLSCALE Procedure"
in SAS Institute Inc., _SUGI Supplemental Library User's Guide, Version 5
Edition_, Cary, NC: SAS Institute Inc.

Schiffman, S.S., Reynolds, M.L., and Young, F.W. (1981), _Introduction
to Multidimensional Scaling_, New York: Academic Press.

Torgerson, W.S. (1958) _Theory and Methods of Scaling_, New York: John Wiley & Sons.

Young, F.W. (1982), "Enhancements in ALSCAL-82" in SAS Institute Inc.,
_Proceedings of the Seventh Annual SAS Users Group International
Conference_, Cary, NC: SAS Institute Inc.

Young, F.W. (1987), _Multidimensional Scaling: History, Theory, and
Applications_, edited by R.M. Hamer. Hillsdale, NJ: Lawrence Erlbaum
Associates.

Young, F.W., Lewyckyj, R., and Takane, Y. (1986), "The ALSCAL Procedure"
in SAS Institute Inc., _SUGI Supplemental Library User's Guide, Version 5
Edition_, Cary, NC: SAS Institute Inc.

-- 

Warren S. Sarle       SAS Institute Inc.   The opinions expressed here
saswss@unx.sas.com    SAS Campus Drive     are mine and not necessarily
(919) 677-8000        Cary, NC 27513, USA  those of SAS Institute.
