One-to-one correspondences in pyramidal representation: A unified approach. (English) Zbl 0733.92029

Durand, Carole; Fichet, Bernard

Classification and related methods of data analysis, Proc. 1st Conf. IFCS, Aachen/FRG 1987, 85-90 (1988).

Summary: [For the entire collection see Zbl 0732.00020.]
Recently, two extensions of Johnson’s theorem [S. C. Johnson, Psychometrika 32, 241–254 (1967; Zbl 1367.62191)] have been investigated in view of pyramidal representations [see E. Diday, RAIRO, Autom. Prod. Inf. Ind. 20, 475–526 (1986; Zbl 0592.62052)]. These extensions establish two bijections between particular Robinsonian dissimilarities and (weakly) indexed pseudo-hierarchies. Using the concept of balls, a simpler proof is here given for a more general theorem displaying the successive generalizations. Finally, some approximation problems are discussed.

MSC:

91C20

Clustering in the social and behavioral sciences

Keywords:

extensions of Johnson’s theorem; pyramidal representations; Robinsonian dissimilarities; indexed pseudo-hierarchies; balls

Citations:

Zbl 0732.00020; Zbl 0592.62052; Zbl 1367.62191