[Forum SIS] Seminar: okada a DISMEQ Department of Statistics and Quantitative Methods, University of Milano-Bicocca

Lun 18 Nov 2019 08:27:05 CET

-----------------------------------------------------------------------------------
**   21/11/2019  h.11.00 /Building U7  4th floor - Seminar Room 4026/

    Department of Statistics and Quantitative-Methods,
        Via Bicocca degli Arcimboldi, 8 - 20126 Milano

-----------------------------------------------------------------------------

   MULTIDIMENSIONAL SCALING OF ASYMMETRIC RELATIONSHIPS

  Akinori Okada
  Professor Emeritus of the Rikkyo University, Tokyo
okada a rikkyo.ac.jp

Torgerson (1952) introduces his multidimensional scaling as the first 
practical method of representing proximity (similarity, dissimilarity) 
relationships among objects in a multidimensional space or a 
configuration. The method is called metric, which means that the 
observed proximities are assumed to be interval or ratio scaled values. 
Shepard (1962a, b) and Kruskal (1964a, b) introduced methods which are 
called nonmetric multidimensional scaling. Nonmetric multidimensional 
scaling assumes that the observed proximities are ordinal scaled values. 
This mean that only the information which of two proximities is larger 
(or smaller) than the other is utilized in the analysis. The nonmetric 
multidimensional scaling of Kruskal (1964a, b) has been used most 
frequently in practical applications of multidimensional scaling, and 
several multidimensional scaling methods based on the similar idea of 
Kruskal (1964a, b) have been introduced. The multidimensional scaling 
developed so far assumes that the proximity is symmetric.

Several researchers focused their attention on the asymmetry in various 
sorts of proximity; similarity/dissimilarity judgment, same/different 
judgment, reaction time, social and psychological distance, and so on. 
It is meaningful to develop asymmetric multidimensional scaling which 
can represent asymmetric proximity relationships among objects in a 
configuration. Various kinds of procedures to analyze an asymmetric 
proximity matrix were invented. We concentrate on the multidimensional 
scaling which is based on the distance among points, each representing 
an object, in a multidimensional space (distance model).

The asymmetric multidimensional scaling by Okada and Imaizumi (1987) 
represents asymmetric proximity relationships among objects by the 
interpoint distance in a multidimensional configuration. The distance 
between two points is symmetric. It is difficult to represent asymmetric 
relationships simply by the interpoint distance, because the distance 
itself is symmetric. The model of Okada and Imaizumi (1987), which is 
called Distance-radius model, represents an object by a point and a 
circle [two-dimensional model] (sphere [three-dimensional] or 
hypersphere [larger than three-dimensional]) centered at the point in a 
multidimensional space. Several models were derived from the model.

Multidimensional scaling mentioned above, regardless of metric or 
nonmetric and of being able to represent asymmetry or not, analyzes one 
proximity matrix (object x object) or one-mode two-way proximities 
(Carroll and Arabie, 1980). To analyze a set of proximity matrices each 
comes from a source (object x object x source) or two-mode three-way 
proximities, INDSCAL was developed by Carroll and Chang (1970). INDSCAL 
can represent relationships among objects and how each source 
(individual) has different relationships among objects each other. While 
INDSCAL can represents differences among sources, it cannot represent 
the asymmetry in relationships among objects of each source. The model 
of Okada and Imaizumi (1997) is an extension of the Distance-radius 
model which can analyze a set of asymmetric proximity matrices or 
two-mode three-way asymmetric proximities. The asymmetric 
multidimensional scaling of Okada and Imaizumi (1997) can represents 
asymmetric relationships among objects, and how each source has 
different symmetric and asymmetric relationships among objects each 
other as well. Several models, which are less restricted or more 
restricted, have been derived from Okada and Imaizumi (1997).

To analyze one-mode two-way asymmetric proximities, there is another 
asymmetric multidimensional scaling by Okada (2011) and Okada and 
Tsurumi (2012) which is based on the singular value decomposition. The 
asymmetric multidimensional scaling has been utilized to analyze mainly 
the data from marketing research or brand switching. Several models and 
procedures have been derived by extending the model of Okada (2011) 
which can, for example, evaluate newly introduced brands, diagnose 
brands, tell how to increase the dominance of a brand in the brand 
switching... Another sort of procedure to analyze one-mode two-way 
asymmetric proximities is asymmetric cluster analysis, which can 
represent asymmetric proximity relationships among objects by forming 
clusters (Okada & Iwamoto, 1966; Okada & Yokoyama, 2015). It is 
important and meaningful to develop not only multidimensional scaling 
but also cluster analysis as well so that we can use two different kinds 
of methods to analyze the same proximity data (Arabie & Hubert, 1994).

There are quite a few number of applications of asymmetric 
multidimensional scaling and cluster analysis. While these applications 
are not mentioned here because of the space limitation, some of them 
will be mentioned in the talk. The present abstract deals only with 
researches of the author and his colleagues. Bove and Okada (2018) gives 
a review which covers a wide range of asymmetric multidimensional 
scaling and asymmetric cluster analysis in detail. Some books mention 
the analysis of asymmetric relationships. Borg and Groenen (2005) has a 
chapter of asymmetric multidimensional scaling. Coxon and Coxon (2001) 
refers to asymmetric multidimensional scaling.

**/References///

Arabie, P.  & Hubert, L. (1994). Cluster analysis in marketing research. 
In Bagozzi, R. F. (Ed.), A/dvanced methods marketing research/. pp. 
160-199. Cambridge, MA: Blackwell.

   Bove, G., & Okada, A. (2018). Methods for the analysis of asymmetric 
pairwise relationships. Advances in Data Analysis and Classification, 
*12*, 5-31.

     Borg, I., & Groenen, P.J.F. (2005). /Modern multidimensional 
scaling: Theory and applications/(2nd ed.). New York: Springer.

     Carroll, J. D., & Arabie, P. (1980). Multidimensional Scaling. In 
M. R. Rosenzweig & L. W. Porter (Eds.), /Annual Review of Psychology/, 
*31*, 607-649.

     Carroll, J. D. & Chang, J. J. (1970). Analysis of individual 
differences individual differences in multidimensional scaling via an 
/N/-way generalization of “Eckart-Young” decomposition. /Psychometrika/, 
/35/, 283-319.

     Coxon, T.F., & Coxon, M.A.A. (2001). /Multidimensional scaling/ 
(2nd ed.). Boca Raton, FLChapman&Hall/CRC.

      Kruskal, J. B. (1964a). Nonmetric multidimensional scaling by 
optimizing goodness of fit to a nonmetric hypothesis. /Psychometrika/, 
/29/, 1-27.

      Kruskal, J. B. (1964b). Nonmetric multidimensional scaling: A 
numerical method. /Psychometrika/, *29,* 115-129.

    Okada, A. (2011). Analyzing social affinity for foreigners by 
spectral decomposition: Asymmetric multidimensional scaling of EASS 2008 
Data [Supekutoru bunkai niyoru gaikokujin ni taisuru shinkinkan no 
bunseki: EASS 2008 no deta wo mochiita hitaishou tajigenshakudokoueihou 
no ouyou]. In JGSS Research Center, (Ed.), /JGSS Research Series/, /11/. 
(pp. 119-128). Osaka: Osaka University of Commerce.

      Okada, A., & Imaizumi, T. (1987). Nonmetric multidimensional 
scaling of asymmetric proximities. /Behaviormetrika, /*21*/,/ 81-96.

     Okada, A., & Imaizumi, T. (1997). Asymmetric multidimensional 
scaling of two-mode three-way proximities. /Journal of 
Classification/,*14, *195-224.

    Okada, A., & Iwamoto, T. (1996). University enrollment flow among 
the Japanese prefectures: A comparison before and after the Joint First 
Stage Achievement Test by asymmetric cluster analysis. 
/Behaviormetrika/, *23*, 169-185.

    Okada, A., & Tsurumi, H. (2012). Asymmetric multidimensional scaling 
of brand switching among margarine brands./Behaviormetrika/, *39*, 111-126.

     Okada, A., & Yokoyama, S. (2015). Asymmetric CLUster analysis based 
on SKEW-Symmetry: ACLUSKEW. In I. Morlini, M. Tommaso, & M. Vichi, 
(Eds.), /Advance in statistical models for data analysis/.//191-199. 
Heidelberg, Germany: Springer-Verlag.

     Shepard, R. N. (1963a). The analysis of proximities: 
Multidimensional scaling with unknown distance function. I. 
/Psychometrika/, *27*, 125-140.

     Shepard, R. N. (1963b). The analysis of proximities: 
Multidimensional scaling with unknown distance function. II. 
/Psychometrika/, *27,* 219-246.

       Torgerson, W. T. (1952). Multidimensional scaling: I. Theory and 
method. /Psychometrika/, *17*, 401-419.

**

/For more info:/***//**/fulvia.pennoni a unimib.it 
<mailto:fulvia.pennoni a unimib.it>/*

-- 
Fulvia Pennoni Ph.D.
Department of Statistics and Quantitative Methods
University of Milano-Bicocca
https://sites.google.com/view/fulviapennoni

-------------- parte successiva --------------
Un allegato HTML è stato rimosso...
URL: <http://www.stat.unipg.it/pipermail/sis/attachments/20191118/6c96bbb6/attachment-0001.html>