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\begin{document}
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 \title{\huge{A complex analysis for two-mode three-way 
       asymmetric relational data} \\
     \vspace*{5cm} }
        \author{Naohito Chino \\
                Aichi Gakuin University \\
                Axel Grorud \\
                Universit\'e de Provence \\
                Ryozo Yoshino \\
                The Institute of Statistical Mathematics \\
            \vspace{5cm} \\
  Handout presented at the Fifth Conference of the International \\
       Federation of Classification Societies, Kobe, Japan \\}
%
%\begin{center}
%  Naohito Chino$^1$, Axel Grorud$^2$ and Ryozo Yoshino$^3$ \\[0.3cm]
%
%{\small $^1$ Aichi Gakuin University\\
%             12 Araike, Nisshin-city\\
%             Aichi 470-11, Japan} \\[0.3cm]
%{\small $^2$ Universit\'e de Provence\\
%             39 rue F. Joliot-Curie, 39 rue F. \\
%             Joliot-Curie 13453 Marseille \\ 
%              cedex 13, France} \\[0.3cm]
%{\small $^3$ The Institute of Statistical Mathematics\\
%             Minami-Azabu, Minato-ku\\
%             Tokyo 106, Japan}
%\end{center}
%
  \date{March 29, 1996}
%
\maketitle
%
\newpage
%
{\small\bf Summary:} {\small 
  This paper presents a statistical model of complex data analysis for two-mode
three-way asymmetric relational data.  This model is a generalization of STATIS
(Gla\c con, 1981; Lavit, 1985, 1988; Lechevallier, 1987; L'Hermier des 
Plantes, 1976), therefore it is called a generalized STATIS (GSTATIS).  
For the demonstration of feasibility and stability of GSTATIS, we applied 
it to social survey data, and results showed that the model was valid
to clarify the spatial and temporal structures of objects in the data
sets.}\\[20pt]
%
{\large\bf 1. Introduction}\\[10pt]
 \hspace*{5.8mm}
  A number of methods have been proposed for the analysis of three-way 
data, i.e., $N$ by $m$ data matrices,$\mbC_1, \mbC_2,\cdots,\mbC_q$,
or $m$ by $m$ cross-product matrices, $\mbS_1, \mbS_2,\cdots,$ $\mbS_q$.  
Kiers (1991) considered them as variants of principal component analysis 
(PCA) and examined their hierarchy from an algorithmic point of view. \par
  However, there are few who proposed efficient algorithms for the analysis
of three-way data of square asymmetric matrices, $\mbA_1,\mbA_2,
\cdots,\mbA_q$, although there are several models for these asymmetric 
three-way data (Chino, 1980; Harshman, 1978; Harshman, Green, Wind, \& Lundy,
1982; Kiers, 1995; Okada \& Imaizumi, 1992, 1995; Young \& Lewyckyj, 1979;
Zielman, 1991). \par
    Recently, however, an efficient algorithm for the analysis of such a
set of asymmetric matrices has been proposed by Grorud, Chino, and Yoshino
(1995) (for more detail, see Chino, Grorud, \& Yoshino, 1996).  This is a
natural generalization of STATIS ({\em S}tructuration des {\em TA}bleaux 
\`a {\em T}rois {\em I}ndices de la {\em S}tatistique), and is called the
{\em Generalized STATIS} (GSTATIS).  A prototype of STATIS was originally
proposed by L'Hermier des Plantes (1976), and developed further by Gla\c con 
(1981), Lavit (1985, 1988), and Lechevallier (1987). \par
    In this paper we show some examples of application of GSTATIS to social
survey data to demonstrate the feasibility and stability of it. \\ [15pt]
%
{\large\bf 2. Three steps of GSTATIS}\\[10pt]
 \hspace*{5.8mm}
  The algorithm of GSTATIS consists of three steps, i.e., the inter-structure
analysis, the intra-structure analysis, and the trajectory analysis, as in
STATIS. \par
  A major difference between STATIS and GSTATIS is in the data matrices.
That is, GSTATIS assumes that these data are asymmetric (dis-)similarity
matrices at $q$ occasions {\em measured at a ratio scale level}, whereas 
STATIS assumes that those are either the $N$ by $m$ data matrices, $\mbX_1,
\mbX_2,\cdots,\mbX_q$, or the $m$ by $m$ cross-product matrices computed 
from them. \par
  The second major difference is that STATIS examines the structures of 
objects and occasions in a real number space, whereas GSTATIS deals with
those in a complex number space.  The extension of number space to the 
complex space clarifies metric structures of asymmetric matrices which
cannot be done in the real number space.\par
  GSTATIS requires coordinates of objects at each occasion before we
proceed to the three-step analysis.  To obtain these coordinates, we 
analyze each of the asymmetric relational data matrices, $\mbA_1,\mbA_2,
\cdots,\mbA_q$, via the {\em Hermitian Form Model} (HFM) for the analysis
of asymmetry (Chino, 1991; Chino \& Shiraiwa, 1993; Escoufier \& Grorud, 
1980) and examine the complex space structures contained in each of them.  
HFM is nothing but an eigenvalue-eigenvector decomposition of the Hermitian 
matrix $\mbH$ constructed from certain square asymmetric matrix $\mbA$ in 
such a way that $\mbH=\mbA_s + \mbA_{sk}$.  Here, $\mbA_s$ and $\mbA_{sk}$ 
are, respectively, the symmetric part and the skew-symmetric part of $\mbA$.
Then, to eliminate the local rotational indeterminacy, we rotate each of
the complex dimensions of each of the configurations into the planar
principal axes. \par 
  In Step 1, GSTATIS computes a generalized RV coefficient between the 
Hermitian matrices at occasion $k$ and occasion $l$ defined by
%
%                                                         eq. (1)
%
\begin{eq} RV(\mbH_k,\,\mbH_l)=tr(\mbH_k \mbH_l)/\left\{tr{(\mbH_k)}^2
            tr{(\mbH_l)}^2 \right\}^{1/2}.
\end{eq}
%
Here, $\mbH_k$ and $\mbH_l$ are Hermitian matrices computed from data
matrices $\mbA_k$ and $\mbA_l$, respectively. \par
    In Step 2, GSTATIS computes a weighted sum of Hermitian matrices, $\mbH_1,
\mbH_2,\cdots,$ $\mbH_q$, whose weights are the elements of the eigenvector
corresponding to the largest eigenvalue of the RV coefficient matrix computed 
in Step 1.  It should be noticed that, in this case, the eigenvector is
real.  Let the eigenvector be $\mbbeta$=${(\beta_1,\beta_2,\cdots,
\beta_q)}^t$.  Then, the weighted sum matrix $\mbV$ is written as
%
%                                                           eq. (2)
\begin{eq}  \mbV=\sum_{k=1}^q \,\beta_k\,\mbH_k.  \end{eq}
%
%
\begin{table}[htb]
\caption[table1]{Eigenvalues of the empirical Hermitian
                 matrix of the first data in their order.}
%\caption[table1]{}
\label{Eigenvalues}
\begin{center}
 \begin{tabular}{|c|ccccccc|} \hline
  order$\backslash$nation & JP & FRG & FR & UK & USA & NL & IT \\
 \hline
 1 & 24.8 & 27.5 & 17.8 & 28.2 & 23.3 & 26.7 & 26.6 \\
 2 & 17.3 &  8.4 &  7.9 & 11.2 & 13.9 & 14.5 &  4.8 \\
 3 &  3.6 &  2.8 &  4.4 &  0.7 &  2.0 &  0.7 &  0.9 \\
 4 &  1.5 &  0.8 &  2.1 &  0.1 &  1.1 &  0.3 &  0.3 \\
 5 &  0.4 &  0.2 &  1.3 & -0.1 &  0.2 &  0.1 &  0.2 \\
 6 &  0.0 &  0.0 &  0.0 &  0.0 &  0.0 &  0.0 &  0.0 \\
  \hline
 \end{tabular}
\end{center}
\end{table}
%
\par
  It is easy to show that this type of matrix is also Hermitian (see,
Debnath \& Mikusi\'nski, 1990).  Therefore, HFM (thus, PCA of Hermitian
matrix) of the above matrix $\mbV$ enables us to uncover a
multidimensional complex intra-structure among objects, i.e., a 
finite-dimensional complex (f.d.c.) Hilbert space structure or an indefinite 
metric structure. It should
be noticed that we must rotate each of the complex dimensions into the
planar principal axes, as is the case of the preparatory step. \par
    In Step 3, GSTATIS projects the complex multidimensional structure of
objects at each occasion, which is obtained by HFM of each of the
Hermitian matrices, onto the holistic intra-structure among
objects disclosed in step 2, to yield the trajectory of each object
through occasions in the holistic structure.  \par
  Suppose that the complex coordinate vector of objects on dimension $t$
at occasion $k$ is $\mby_{tk}$.  Moreover, let the matrix $\mbU_p$ be composed
of the $p$ eigenvalues of the matrix $\mbV$ in step 2.  Then it is easy to
prove that the {\em orthogonal projector} $\mbQ$ can be written as
%
%                                                           eq. (3)
%
\begin{eq} \mbQ=\mbU_p\,{(\mbU_p^*\,\mbU_p)}^{-1}\mbU_p^*,\end{eq}
%
where $\mbU_p^*$ is the {\em conjugate transpose} of $\mbU_p$ (Lancaster \&
Tismenetsky, 1985). \par
  Using $\mbQ$, we can define $\mbt_{tk}=\mbQ\,\mby_{tk}$.  The vectors,
$\mbt_{tk},\,k=1,\cdots,q$, constitute the trajectory discussed above.
In this case, however, it should be noticed that these vectors are not real
but {\em complex} in general.  As is the case for STATIS, the term
${(\mbU_p^*\,\mbU_p)}^{-1}$ of (3) is equal to the identity matrix $\mbI_p$. 
\par
%
%  Figure 1
%
 \begin{figure}[hbt]
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 \begin{picture}(130,100) (-11,0)
%
% Figure 1-a
%
 \put(0,50){\dashbox{1}(65,50){}}
 \put(5,78){\vector(1,0){50}} % horizontal axis
 \put(24.5,54){\vector(0,1){43}} % vertical axis
%
 \put(21,97){$imag$}
 \put(56,78){$real$}
 \put(36,52){$1-a.\;France$}
%
 \put(47,89){$CS$}
 \put(33,61){$SS$}
 \put(18,67){$N$}
 \put(14,81.5){$SD$}
 \put(14,88.5){$CD$}
 \put(15.5,85){$DK$}
%
% Figure 1-b
%
 \put(65,50){\dashbox{1}(65,50){}}
 \put(72,74){\vector(1,0){50}} % horizontal axis
 \put(90,55){\vector(0,1){40}} % vertical axis
%
 \put(85,97){$imag$}
 \put(118,75){$real$}
 \put(101,52){$1-b.\;Italy$}
%
 \put(94.5,90){$CS$}
 \put(117,68){$SS$}
 \put(85,65){$N$}
 \put(81,68){$SD$}
 \put(79,71){$CD$}
 \put(78,75){$DK$}
%
% Figure 1-c
%
 \put(0,0){\dashbox{1}(65,50){}}
 \put(9,25){\vector(1,0){50}} % horizontal axis
 \put(24.5,5){\vector(0,1){40}} % vertical axis
%
 \put(21,46){$imag$}
 \put(53,22){$real$}
 \put(36,2){$1-c.\;Holland$}
%
 \put(43,39){$CS$}
 \put(41,8){$SS$}
 \put(17.5,22){$N$}
 \put(11,22){$SD$}
 \put(11,26){$CD$}
 \put(17.5,26){$DK$}
%
% Figure 1-d
%
 \put(65,0){\dashbox{1}(65,50){}}
 \put(74,31){\vector(1,0){48}} % horizontal axis
 \put(90,2){\vector(0,1){47}} % vertical axis
%
 \put(91,46){$imag$}
 \put(118,26){$real$}
 \put(101,2){$1-c.\;Japan$}
%
 \put(113,42){$CS$}
 \put(95,7){$SS$}
 \put(78.5,30){$N$}
 \put(76.5,34){$SD$}
 \put(83,37){$CD$}
 \put(82,33){$DK$}
%
 \end{picture}
%
\caption[figure.1]{Preparatory HFM analyses of the first data set.}
%\caption[figure.1]{}
 \label{fig:fig1}
%
 \end{figure}
%
  However, if we may introduce a diagonal weight-metric in the eigen-space
of the holistic space, the above term does not necessarily vanish. 
That is,
%                                                             eq. (4)
\begin{eq}
   \mbQ=\mbD^{\frac12}\mbU_p\,{(\mbU_p^* \mbD \mbU_p)}^{-1}
                                     \mbU_p^* \mbD^{\frac12}.
\end{eq}
%
%
{\large\bf 3. Result}\\[10pt]
%
\hspace*{5.8mm}
%
    Two sets of data were obtained by computing cross-classification tables
of some questionnaire items of the international survey data conducted by
The Institute of Statistical Mathematics in Japan (Hayashi, Suzuki, \& 
Leghorn, 1991; Hayashi, Suzuki, \& Sasaki, 1992; Yoshino, 1992) for 7 
developed countries, Japan, FRG, France, UK, USA, the Netherlands, and Italy.  
Face to face interviews were used, and respondents were randomly chosen in 
the nationwide sampling at each country.  Sample sizes were, respectively, 
2265, 1000, 1013, 1043, 1563, 1083, and 1048.  Considering the differences 
in size, we used the percentages of the original frequencies.  \par
  In the first set of data the row and the column questionnaire items, 
respectively, in each of the tables were as follows: (Q 28) say, for Americans,
"All things considered, how satisfied are you with your family life, --the 
time you spend and the things you do with members of your family?  Just call
off the letter which comes closest to your feelings, (Q 29) Now I  want to ask 
about your life as a whole.  How satisfied are you with your life as a whole
these days?  Which letter on this card comes closest to your feeling? \par
%
%  Figure 2
%
 \begin{figure}[hbt]
 \setlength{\unitlength}{1mm}
 \begin{picture}(100,100) (-26,0)
%
 \put(10,62.5){\vector(1,0){80}} % horizontal axis
 \put(50,9){\vector(0,1){80}} % vertical axis
%
 \put(45,93){$Dim.3$}
 \put(86,54){$Dim.2$}
%
 \put(62,77){$Japan$}
 \put(27,64){$FRG$}
 \put(68,17){$France$}
 \put(48.5,75){$UK$}
 \put(55,74){$USA$}
 \put(55,82){$Holland$}
 \put(25,44){$Italy$}
%
 \end{picture}
%
\caption[figure.2]{Inter-structure analysis of the first data set.}
%\caption[figure.2]{}
 \label{fig:fig2}
%
 \end{figure}
%
  The rating scale categories for these items were (1) "Completely satisfied"
(CS), (2) "Somewhat satisfied" (SS), (3) "Neither completely satisfied nor 
completely dissatisfied (neutral)" (N), (4) "Somewhat dissatisfied" (SD), 
(5) "Completely dissatisfied" (CD), (6) "Other" or "Don't know" (DK).  \par
  Before we proceed to the GSTATIS analysis, it will be interesting to 
analyze each of the seven Hermitian matrices of the first set of data for
the seven nations by HFM, and examine the complex metric structures contained
in them.  Table 1 shows eigenvalues for each of the matrices in their order.  
We can conclude that all the matrices have f.d.c. Hilbert space structures, 
since all the eigenvalues of these matrices except for UK are positive, and the
fifth eigenvalue with negative sign for UK may be negligible.  The sixth 
eigenvalue of zero value for each nation is due to the double centering
transformation of the percentage (frequency) data.  \par
  Figure 1 shows configurations of categories for four nations, France,
Italy, Holland, and Japan out of seven on dimension 1 via HFM.  In HFM, 
each dimension represents a complex plane, and thus the horizontal axis 
and the vertical axis correspond to real and imaginary axes, respectively. \par
  Considering the sign of the largest eigenvalue corresponding to Dimension 1
for each nation, and the magnitudes of the parallelogram spanned by position
vectors of six categories, all of the four planes in Figure 1 indicate a
salient asymmetric relation from CS to SS.  This means that some people are
not fully satisfied with their life as a whole, although they are completely
satisfied with their family life in these nations.  This tendency was 
common to all the seven nations.  However, it is also apparent from these
configurations that there exist delicate shades of characteristics among
these configurations.  GSTATIS may enable to extract major factors of such 
differences in characteristics.
%
%  Figure 3
%
 \begin{figure}[hbt]
 \setlength{\unitlength}{1mm}
 \begin{picture}(100,100) (-36,0)
%
 \put(2,35){\vector(1,0){75}} % horizontal axis
 \put(27,2){\vector(0,1){75}} % vertical axis
%
 \put(16,73){$imag$}
 \put(69,36){$real$}
%
 \put(49,70){$CS$}
 \put(70,12){$SS$}
 \put(15,23){$N$}
 \put(7,30){$SD$}
 \put(8,41){$CD$}
 \put(7,37){$DK$}
%
 \end{picture}
%
\caption[figure.3]{Intra-structure analysis of the first data set.}
%\caption[figure.2]{}
 \label{fig:fig3}
%
 \end{figure}
%
\par
  Figure 2 shows the inter-structure analysis of GSTATIS for the first
data set.  We neglected the first dimension because it was interpreted as
a sort of size factor in principal component analysis.  Dimension 2 contrasts
France with Italy, whereas Dimension 3 Holland with France. In general, it is 
helpful to compare configurations of objects obtained by the repeated 
applications of HFM to Hermitian matrices in interpreting these dimensions.\par
  Comparing configurations shown in Figure 1, dimension 2 may be interpreted 
as amount of people who are completely satisfied with their life as a whole, 
although they are completely dissatisfied with their family life.  
  In the similar way, Dimension 3 may be considered as amount of people who 
are somewhat dissatisfied with their life as a whole, although neutral with 
their family life. \par
%
%  Figure 4
%
 \begin{figure}[hbt]
 \setlength{\unitlength}{1mm}
 \begin{picture}(130,60) (-12,0)
%
% Figure 4-a
%
 \put(0,0){\dashbox{1}(65,60){}}
 \put(3,8){\vector(1,0){55}} % horizontal axis
 \put(12,9){\vector(0,1){42}} % vertical axis
%
 \put(9,53){$imag$}
 \put(53,10){$real$}
 \put(27,2){$4-a.\;Category, CS$}
%
 \put(31,43){CS}
 \put(51,27){$Japan$}
 \put(11,30){$FRG$}
 \put(49,30){$France$}
 \put(39,33){$UK$}
 \put(40,40){$USA$}
 \put(42,37){$Holland$}
 \put(19,40){$Italy$}
%
% Figure 4-b
%
 \put(65,0){\dashbox{1}(65,60){}}
 \put(71,54){\vector(1,0){52}} % horizontal axis
 \put(73,6){\vector(0,1){50}} % vertical axis
%
 \put(68,56.5){$imag$}
 \put(116,55){$real$}
 \put(92,2){$4-b.\;Category, SS$}
%
 \put(110,34){SS}
 \put(82,15){$Japan$}
 \put(120,50){$FRG$}
 \put(87,25){$France$}
 \put(103,30){$UK$}
 \put(104,24){$USA$}
 \put(100,27){$Holland$}
 \put(117,43){$Italy$}
%
 \end{picture}
%
\caption[figure.4]{Trajectories of two categories on dimension 1.}
%\caption[figure.4]{}
 \label{fig:fig4}
%
 \end{figure}
%
%  Figure 5
%
 \begin{figure}[hbt]
 \setlength{\unitlength}{1mm}
 \begin{picture}(150,120) (-5,0)
%
 \put(15,65){\vector(1,0){120}} % horizontal axis
 \put(75,10){\vector(0,1){100}} % vertical axis
%
 \put(69,112){$Dim.\;3$}
 \put(132,57){$Dim.\;2$}
%
 \put(98,16){$Japan$}
 \put(89.5,46){$FRG$}
 \put(22.5,31){$France$}
 \put(77,91){$UK$}
 \put(63,98){$USA$}
 \put(72,68){$Netherlands$}
 \put(71,88){$Italy$}
%
 \end{picture}
%
 \caption[figure.5]{Inter-structure analysis of the second data.}
 \label{fig:fig5}
%
 \end{figure}
%
  Figure 3 shows the first dimension of the intra-structure analysis of
the first data set.  As is the case with the HFM analysis, each of the
dimensions in the intra-structure analysis represents a complex plane.
Therefore, interpretation of each of them may be done in the similar manner
as that of HFM.  Figure 3 may be viewed as an average configuration of the
configurations of the seven nations via HFM, some of which have already been
shown in Figure 1. \par
  Figure 4 shows two of the trajectories obtained by the trajectory analysis
of GSTATIS for the first set of data.  Although these trajectories are 
obtained per dimension as well as per object, it is sufficient for us to
pick up a few of them whose variations of coordinates are relatively large
in order to examine the major characteristics of variations of occasions
in the holistic space, i.e., the configuration of objects obtained via the 
intra-structure analysis. \par
  Figure 4-a depicts the trajectory of occasions, i.e., nations, and the
position of Category CS, i.e., "Completely satisfied", in the configuration
of dimension 1 of the holistic space.  In the similar manner, Figure 4-b 
draws the trajectory of nations and the position of Category SS, i.e.,
"Somewhat satisfied" in the configuration of dimension 1 of the space.
These trajectories generally make clear the differences in occasions of 
the holistic structure of objects. \par
  Finally, we shall show the result of the GSTATIS analysis for the second
data set.  For space limitation, we shall present only the result of the
inter-structure analysis.  The row and the column questionnaire items, 
respectively, in each of the
tables in the second set of data were as follows:  (Q1) say, for Americans,
"Compared with ten years ago do you think the standard of living of
Americans as a whole is ...?", (Q2) "Compared with
ten years ago do you think your standard of living is ...?".  \par
  The rating scale categories for these items were (1) "Much better", (2)
"Slightly better", (3) "About the same", (4) "Slightly worse, or",
(5) "Much worse", and (6) "Don't know". \par
  Figure 5 shows the result of the inter-structure analysis for the second 
set of data.  Dimension 2 shown in this figure can be interpreted intuitively 
as a measure which distinguishes the recognitions about standard of living 
of the Japanese from those of the French.  \par
  Comparison of the two configurations of objects obtained by HFM leads to 
the following curious conclusion:  On the one hand, compared with ten years 
ago individual Japanese standards of living are not perceived to have risen 
relative to the rise in the average national standard of living.  On the 
other hand, the average French national standard of living is not perceived 
to have risen relative to the rise in their individual standards of living.  
This result was validated also in the Steps 2 and 3 analyses (for more detail,
see Chino, Grorud, \& Yoshino, 1996).
\\[15pt]
%
%\newpage
%
\begin{center}
{\large\bf References}
\end{center}
%\\
\par
%
\vskip1mm
\everypar{\hangafter=1\hangindent=.5cm}\parindent=0cm
%
Chino, N. (1980). A unified geometrical interpretation of the MDS techniques
  for the analysis of asymmetry and related techniques.  {\em Paper
  presented at the symposium on "Asymmetric multidimensional scaling" at
  the Spring Meeting of the Psychometric Society}, Iowa, U.S.A. \par
Chino, N. (1991).  A critical review for the analysis of asymmetric
  relational data.  {\em Bulletin of The Faculty of Letters of Aichi Gakuin
  University}, {\bf 21}, 31-52.  \par
Chino, N., Grorud, A., \& Yoshino, R. (1996).  A complex data analysis for
  two-mode three-way asymmetric relational data.  {\em Paper submitted for
  publication}. \par
Chino, N., \& Shiraiwa, K. (1993).  Geometrical structures of some
  non-distance models for asymmetric MDS. {\em Behaviormetrika}, {\bf 20},
  35-47. \par
Debnath, L., \& Mikusi\'nski, P. (1990).  {\em Introduction to Hilbert Spaces
  with Applications}. New York: Academic Press. \par
Escoufier, Y., \& Grorud, A. (1980). Analyse factorielle des matrices
  carrees non symetriques.  In Diday, E. (Eds.), {\em Data Analysis and
  Informatics} (pp.263-276). New York: North Holland.  \par
Gla\c con, F. (1981).  {\em Analyse conjointe de plusieurs matrices
  de donn\'ees.}  Unpublished doctoral dissertation, University of Grenoble,
  France. \par
Grorud, A., Chino, N., \& Yoshino, R. (1995).  Complex analysis for three-way
  asymmetric relational data.  {\em Proceedings of the 23th annual meeting of
  the Behaviormetric Society of Japan}, Osaka, Japan. \par
Harshman, R. A. (1978).  Models for analysis of asymmetrical relationships
  among $N$ objects or stimuli.  {\em Paper presented at the First Joint
  Meeting of the Psychometric Society and The Society for Mathematical
  Psychology}, Hamilton, Canada. \par
Harshman, R. A., Green, P. E., Wind, Y., \& Lundy, M. E. (1982).  A model for
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