Abstract
Formulations that specify the coordinates of multiple objects in multidimensional space, where the dissimilarities among objects do not satisfy the axioms of distance, are presented in this study. It is shown that dissimilarities not satisfying triangle inequality can be treated by extending classical multidimensional scaling (MDS) to indefinite metric space. It is also shown that asymmetric dissimilarities can be treated by extending classical MDS to complex vector space. Finally, the above two formulations are unified by introducing indefinite metric complex vector space. For numerical calculations, the problem reduces to a matrix completion and is described as semidefinite programming. Graphical representations are also proposed to visualize the numerical results.
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References
Benz, W.: Classical Geometries in Modern Contexts, 3rd edn. pp. 139–141. Springer, Basel (2012)
Berlin T.H., Kac C.: The spherical model of ferromagnet. Phys. Rev. 86, 821–835 (1952)
Borchers B.: CSDP, A C library for semidefinite programming. Optim. Methods Softw. 11, 613–623 (1999)
Borg, I., Groenen, P.: Modern Multidimensional Scaling, 2nd edn. Springer, New York (2005)
Cecil, T.E.: Lie Sphere Geometry, 2nd edn. pp. 39–43. Springer, New York (2008)
Chino N.: A graphical technique for representing the asymmetric relationships between N objects. Behaviormetrika 5, 23–40 (1978)
Chino N.: A generalized inner product model for the analysis of assymetry. Behaviormetrika 27, 25–46 (1990)
Chino N., Shiraiwa K.: Geometrical structures of some non-distance models for asymmetric MDS. Behaviormetrika 20, 35–47 (1993)
Chino N.: A brief survey of asymmetric MDS and some open problems. Behaviormetrika 39, 127–165 (2012)
Escoufier, Y., Grorud, A.: Analyse factorielle des matrices carées non-symétriques. In: Data Analysis and Informatics, pp. 263–276. North-Holland, Amsterdam (1980)
Gower J.C.: Some distance properties of latent root and vector methods used in multivariate analysis. Behaviormetrika 53, 325–338 (1966)
Gower, J.C.: The analysis of asymmetry and orthogonality.: In: Barra, J.R., Brodeau, F., Romer, G., van Cutsem, B. (eds.) Recent Developments in Statistics, pp. 109–123. North-Holland, Amsterdam (1977)
Jolliffe, I.T.: Principal Component Analysis, 2nd edn. pp. 389–390. Springer, New York (2002)
Kiers H.A.L., Takane Y.: A generalization of GIPSCAL of nonsymmetric data. J. Classif. 11, 79–99 (1994)
Krumhansl C.L.: Concerning the applicability of geometric models to similarity data: the interrelationship between similarity and spatial density. Psychol. Rev. 85, 445–463 (1987)
Kumagai A.: Analysis of asymmetric relational data based on classical multidimensional scaling (In Japanese). Trans. Japan Soc. Ind. Appl. Math. 20, 57–66 (2010)
Kumagai A.: Handling asymmetry of relational data from a viewpoint of classical multidimensional scaling (In Japanese). Trans. Japan Soc. Ind. Appl. Math. 21, 165–174 (2011)
Okada A., Imaizumi T.: Nonmetric multidimensional scaling of asymmetric proximities. Behaviormetrika 21, 81–96 (1987)
Saito T., Takeda S.: Multidimensional scaling of asymmetric proximity: model and method. Behaviormetrika 28, 49–80 (1990)
Saito T.: Analysis of asymmetric proximity matrix by a model of distance and additive terms. Behaviormetrika 29, 45–60 (1991)
Saito T., Yadohisa H.: Data Analysis of Asymmetric Structures. Marcel Dekker, New York (2005)
Torgerson W.S.: Theory and Methods of Scaling. Wiley, New York (1958)
Tversky A.: Features of similarity. Psychol. Rev. 84, 327–352 (1977)
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Kumagai, A. Extension of classical MDS to treat dissimilarities not satisfying axioms of distance. Japan J. Indust. Appl. Math. 31, 111–124 (2014). https://doi.org/10.1007/s13160-013-0127-z
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DOI: https://doi.org/10.1007/s13160-013-0127-z