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APPENDIX D: MULTIDIMENSIONAL SCALING TECHNIQUES
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SPATIAL MODEL FOR PERCEPTUAL RELATIONSHIPS
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It has been found most useful to employ  a spatial model to represent
the judged  relationships between sets of stimuli,  such as  auditory
signals.  Multidimensional scaling algorithms have been developed for
the reduction of the very  complex data obtained from the evaluations
of subjective  relationships between all members in a set of stimuli.
The  result  is displayed  in  a  form  which  is  much  more  easily
comprehended and interpreted by the investigator, that of a geometric
configuration of points which  represent the individual stimuli.  The
structure of the subjective evaluations of the set of stimuli is then
mapped into an  n-dimensional space,  where the distances between the
points are determined  by some measure of  the psychological distance
between all pairs of stimuli.

The psychological measures which  can be mapped into a  spatial model
in  terms  of  distance  include  the  confusability  or  the  judged
similarity of stimuli.  The first task involves the identification of
individual signals, perhaps presented under different conditions, and
the  result is a square confusion matrix  of stimulus by response.  A
transform of this matrix yields the relative psychological distances,
directly related to confusability,  of all points in the data matrix.
The second task involves the rating of relative similarity for a pair
of stimuli in the set.  The data can be placed in a  square matrix of
the similarity rating of stimulus i by j, where order of presentation
is preserved.   The psychological distance in this case is  inversely
related to the similarity of stimuli, and is directly related to their
dissimilarity.

The common  characteristic of the scaling programs  we find useful is
their generation  of  an  empirically-based  representation  of   the
relationships  between the stimuli,  rather  than some  theoretically
imposed, a priori representation. We proceed from the perceptual data
and  will compare  the  representation  of  this data  to  the  known
physical attributes of the stimuli.  The uncovering of psychophysical
relationships is a matter of  interpreting the representation of  the
percetual data.   We are concerned  with both the  dimensionality and
the  general properties  of the  space.   Correlations  with physical
parameters are sought.   Various programs exist  which are useful  in
assessing the  correspondence of data  structures,  so that  it would
prove fruitful to formally represent the physical data.
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MULTIDIMENSIONAL SCALING ALGORITHMS
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We  will  briefly  describe the  two  programs  for  multidimensional
scaling found most  useful in our research, MDSCAL and INDSCAL. These
programs both attempt to represent input data matrices in the form of
a  configuration  of points  located  in  an n-dimensional  geometric
space, where n, the number of dimensions, is specifiable by the user.
The  points correspond to the stimuli,  whose psychological distances
are given in the input matrices.  The coordinates  of the  points are
obtained by an iterative computational algorithm which optimizes  the
correspondence of interpoint  distances in the spatial representation
to the measured psychological distances between the stimuli. 

MDSCAL peforms a  non-metric multidimensional  scaling.  The  optimal
spatial  representation of  a subjective  response  matrix is  one in
which the rank order of  the values of psychological distance be  the
same  as  the   rank  order  of  the  interpoint   distances  in  the
n-dimensional  geometric  configuration.  A  monotonic  function maps
psychological values  into distances  in  the spatial  representation
(for a discussion of the theory and procedures of this algorithm  see
Shepard, 1962a, 1962b; Kruskal  1964a, 1964b).  The use  of MDSCAL in
conjunction  with   other  programs  which  deal  with  psychological
distance matrices can be particularly informative.  One such program,
HICLUS,   produces a  tree-structure which represents  the hierarchic
clustering  relationships of stimuli  in the matrix  as inferred from
their psychological distances (the use of HICLUS with  MDSCAL for the
analysis of confusion matrices  for speech signals is demonstrated by
Shepard, 1972).

INDSCAL  is a metric  multidimensional scaling  program,   which  was
developed to  utilize the individual differences  in sets of response
matrices  for  analysis.     It  generates  a   single  n-dimensional
representation  for the complete  set of  matrices.  It  analyzes the
variations  in the  set  of  individual  data  matrices  to  uniquely
determine a  rotation for the  axes in the  space.  Also  produced by
this  analysis is  a representation  of weightings which  account for
individual response  variations  along  the spatial  dimensions.  The
weightings  are mapped  into an  n-dimensional  spatial configuration
which can  be used  to assess  the relationships  between  individual
subjects  or  experimental  conditions  (see  Carroll,  1970,  for  a
complete discussion  of the theory and  operations of this technique;
and Carroll and Wish, **,   for an application of the method for  the
representation of confusion matrices for speech signals).
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