A validation study of a variable weighting algorithm for cluster analysis

De Soete (1986, 1988) proposed a variable weighting procedure when Euclidean distance is used as the dissimilarity measure with an ultrametric hierarchical clustering method. The algorithm produces weighted distances which approximate ultrametric distances as closely as possible in a least squares sense. The present simulation study examined the effectiveness of the De Soete procedure for an applications problem for which it was not originally intended. That is, to determine whether or not the algorithm can be used to reduce the influence of variables which are irrelevant to the clustering present in the data. The simulation study examined the ability of the procedure to recover a variety of known underlying cluster structures. The results indicate that the algorithm is effective in identifying extraneous variables which do not contribute information about the true cluster structure. Weights near 0.0 were typically assigned to such extraneous variables. Furthermore, the variable weighting procedure was not adversely effected by the presence of other forms of error in the data. In general, it is recommended that the variable weighting procedure be used for applied analyses when Euclidean distance is employed with ultrametric hierarchical clustering methods.     

On the use of ordered sets in problems of comparison and consensus of classifications

Ordered set theory provides efficient tools for the problems of comparison and consensus of classifications Here, an overview of results obtained by the ordinal approach is presented Latticial or semilatticial structures of the main sets of classification models are described Many results on partitions are adaptable to dendrograms; many results on n-trees hold in any median semilattice and thus have counterparts on ordered trees and Buneman (phylogenetic) trees For the comparison of classifications, the semimodularity of the ordinal structures involved yields computable least-move metrics based on weighted or unweighted elementary transformations In the unweighted case, these metrics have simple characteristic properties For the consensus of classifications, the constructive, axiomatic, and optimization approaches are considered Natural consensus rules (majoritary, oligarchic, ) have adequate ordinal formalizations A unified presentation of Arrow-like characterization results is given In the cases of n-trees, ordered trees and Buneman trees, the majority rule is a significant example where the three approaches convergeThe authors would like to thank the anonymous referees for helpful suggestions on the first draft of this paper, and W H E Day for his comments and his significant improvements of style     

An error variance approach to two-mode hierarchical clustering

A new agglomerative method is proposed for the simultaneous hierarchical clustering of row and column elements of a two-mode data matrix. The procedure yields a nested sequence of partitions of the union of two sets of entities (modes). A two-mode cluster is defined as the union of subsets of the respective modes. At each step of the agglomerative process, the algorithm merges those clusters whose fusion results in the smallest possible increase in an internal heterogeneity measure. This measure takes into account both the variance within the respective cluster and its centroid effect defined as the squared deviation of its mean from the maximum entry in the input matrix. The procedure optionally yields an overlapping cluster solution by assigning further row and/or column elements to clusters existing at a preselected hierarchical level. Applications to real data sets drawn from consumer research concerning brand-switching behavior and from personality research concerning the interaction of behaviors and situations demonstrate the efficacy of the method at revealing the underlying two-mode similarity structure.     

Hierarchical trees can be perfectly scaled in one dimension

Holman (1972) proved theorems which led him to suggest that there was a fundamental opposition between hierarchical clustering and non-metric Euclidean multidimensional scaling. Empirical experience has shown this to be untrue. Explanations of this apparent contradiction have been offered previously by Kruskal (1977) and Critchley (1986). In this paper we point out the feasibility of perfectly scaling a hierarchical tree in one dimension when the primary approach to ties (Kruskal 1964) is taken. Indeed, there is a whole polyhedral convex cone of solutions for which we obtain an explicit expression.     

Comparison tests for dendrograms: A comparative evaluation

Classifications are generally pictured in the form of hierarchical trees, also called dendrograms. A dendrogram is the graphical representation of an ultrametric (=cophenetic) matrix; so dendrograms can be compared to one another by comparing their cophenetic matrices. Three methods used in testing the correlation between matrices corresponding to dendrograms are evaluated. The three permutational procedures make use of different aspects of the information to compare dendrograms: the Mantel procedure permutes label positions only; the binary tree methods randomize the topology as well; the double-permutation procedure is based on all the information included in a dendrogram, that is: topology, label positions, and cluster heights. Theoretical and empirical investigations of these methods are carried out to evaluate their relative performance. Simulations show that the Mantel test is too conservative when applied to the comparison of dendrograms; the methods of binary tree comparisons do slightly better; only the doublepermutation test provides unbiased type I error. Les arbres utilisés pour illustrés les groupements sont généralement représentés sous la forme de classifications hiérarchiques ou dendrogrammes. Un dendrogramme représente graphiquement l’information contenue dans la matrice ultramétrique (=cophénétique) correspondant à la classification. Dès ultramétriques correspondantes. Nous comparons trois méthodes permettant d’évaluer la signification statistique du coefficient de correlation mesuré entre deux matrices ultramétriques. Ces trois tests par permutations tiennent compte d’aspects différents pour comparer des dendrogrammes: le test de Mantel permute les feuilles de l’arbre, les méthodes pour arbres binaires permutent les feuilles et la topologie, alors que la procédure à double permutation permute les feuilles, la topologie et les niveaux de fusion des dendrogrammes comparés. L’efficacité relative des trois méthodes est évaluée empiriquement et théoriquement. Nos résultats suggèrent l’utilisation préférentielle du test à double permutation pour la comparaison de dendrogrammes: le test de Mantel s’avère trop conservateur, tandis que les méthodes pour arbres binaires ne sont pas toujours adéquates.

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This work was supported by NSERC grant no. A7738 to Pierre Legendre and by a NSERC scholarship to F.-J. Lapointe.     

A new approach to isotonic agglomerative hierarchical clustering

Hierarchical clustering methods must be isotonic for the construction of ultrametric. We present a general strategy to widen the class of isotonic methods implemented by agglomerative algorithms. At each step of the agglomeration we allow one of several admissible pairs to be chosen. Then under mild assumptions an appropriate definition of admissibility guarantees isotony. Moreover we consider the use of the new methods to compute locally optimal ultrametrics. Two examples demonstrate the ability to define new agglomerative methods superior to their traditional competitors.     

Optimal variable weighting for hierarchical clustering: An alternating least-squares algorithm

This paper presents the development of a new methodology which simultaneously estimates in a least-squares fashion both an ultrametric tree and respective variable weightings for profile data that have been converted into (weighted) Euclidean distances. We first review the relevant classification literature on this topic. The new methodology is presented including the alternating least-squares algorithm used to estimate the parameters. The method is applied to a synthetic data set with known structure as a test of its operation. An application of this new methodology to ethnic group rating data is also discussed. Finally, extensions of the procedure to model additive, multiple, and three-way trees are mentioned.The first author is supported as Bevoegdverklaard Navorser of the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek.     

关于逆M-矩阵Hadamard积的一个猜想

对一个n×n逆胙矩阵A,M.Neumann猜想其Hadamard积A°A也是逆M-矩阵.通过许多例子验证,它们都是正确的.迄今为止,猜想未被证出.该文研究了该猜想,给出了一类不同的逆M-矩阵,验证Hadamard积A°A与A°B都是封闭的.进一步验证了猜想:当P≥1,A及任意Ai(i=1,2,…,N-1,N)是逆M-矩阵时,Hadamard幂A°P=(apy),A°∞=(a∞ij),Hadamard积A1°A2°…°AN都是封闭的.     

A graph-theoretic method for organizing overlapping clusters into trees,multiple trees,or extended trees

A clustering that consists of a nested set of clusters may be represented graphically by a tree. In contrast, a clustering that includes non-nested overlapping clusters (sometimes termed a “nonhierarchical” clustering) cannot be represented by a tree. Graphical representations of such non-nested overlapping clusterings are usually complex and difficult to interpret. Carroll and Pruzansky (1975, 1980) suggested representing non-nested clusterings with multiple ultrametric or additive trees. Corter and Tversky (1986) introduced the extended tree (EXTREE) model, which represents a non-nested structure as a tree plus overlapping clusters that are represented by marked segments in the tree. We show here that the problem of finding a nested (i.e., tree-structured) set of clusters in an overlapping clustering can be reformulated as the problem of finding a clique in a graph. Thus, clique-finding algorithms can be used to identify sets of clusters in the solution that can be represented by trees. This formulation provides a means of automatically constructing a multiple tree or extended tree representation of any non-nested clustering. The method, called “clustrees”, is applied to several non-nested overlapping clusterings derived using the MAPCLUS program (Arabie and Carroll 1980).