Extension of classical MDS to treat dissimilarities not satisfying axioms of distance.

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. [ABSTRACT FROM AUTHOR]