Least squares algorithms for constructing constrained ultrametric and additive tree representations of symmetric proximity data

A mathematical programming algorithm is developed for fitting ultrametric or additive trees to proximity data where external constraints are imposed on the topology of the tree. The two procedures minimize a least squares loss function. The method is illustrated on both synthetic and real data. A constrained ultrametric tree analysis was performed on similarities between 32 subjects based on preferences for ten odors, while a constrained additive tree analysis was carried out on some proximity data between kinship terms. Finally, some extensions of the methodology to other tree fitting procedures are mentioned.

[1]  T. Klastorin,et al.  The determination of alternative hospital classifications. , 1981, Health services research.

[2]  G. Soete.  A least squares algorithm for fitting additive trees to proximity data , 1983 .

[3]  W. DeSarbo,et al.  Optimal variable weighting for hierarchical clustering: An alternating least-squares algorithm , 1985 .

[4]  Willem J. M. Levelt,et al.  A scaling approach to the study of syntactic relations , 1970 .

[5]  Reginald G. Golledge,et al.  A heuristic method for the comparison of related structures , 1981 .

[6]  G. Soete.  Additive-tree representations of incomplete dissimilarity data , 1984 .

[7]  G. Soete.  Ultrametric tree representations of incomplete dissimilarity data , 1984 .

[8]  Judith S Reitman,et al.  Organization revealed by recall orders and confirmed by pauses , 1980, Cognitive Psychology.

[9]  Wayne S. DeSarbo,et al.  Constrained classification: The use of a priori information in cluster analysis , 1984 .

[10]  A. D. Gordon.  359. Note: Classification in the Presence of Constraints , 1973 .

[11]  J. Chandon,et al.  Construction de l'ultramétrique la plus proche d'une dissimilarité au sens des moindres carrés , 1980 .

[12]  A. Ferligoj,et al.  Clustering with relational constraint , 1982 .

[13]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[14]  W. DeSarbo,et al.  The representation of three-way proximity data by single and multiple tree structure models , 1984 .

[15]  Vladimir Batagelj,et al.  Some types of clustering with relational constraints , 1983 .

[16]  G. N. Lance,et al.  A General Theory of Classificatory Sorting Strategies: 1. Hierarchical Systems , 1967, Comput. J..

[17]  P A Burrough,et al.  COMPUTER‐BASED SOIL MAPPING OF SMALL AREAS FROM SAMPLE DATA , 1972 .

[18]  Vijay Mahajan,et al.  An Approach to Normative Segmentation , 1978 .

[19]  R. Webster,et al.  COMPUTER‐BASED SOIL MAPPING OF SMALL AREAS FROM SAMPLE DATA , 1972 .

[20]  Ludovic Lebart,et al.  Programme d'agrégation avec contraintes , 1978 .

[21]  George W. Furnas,et al.  The estimation of ultrametric and path length trees from rectangular proximity data , 1984 .

[22]  Geert De Soete,et al.  A least squares algorithm for fitting an ultrametric tree to a dissimilarity matrix , 1984, Pattern Recognit. Lett..

[23]  Robert R. Roediger,et al.  Political redistricting by computer , 1972, CACM.

[24]  C. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[25]  M. J. D. Powell,et al.  Restart procedures for the conjugate gradient method , 1977, Math. Program..

[26]  G. Moore,et al.  Molecular Evolution in the Descent of Man , 1971, Nature.